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Computing the delta set of an affine semigroup: a status report - - PowerPoint PPT Presentation

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Computing the delta set of an affine semigroup: a status report Christopher ONeill San Diego State University cdoneill@sdsu.edu Joint with Thomas Barron* and Roberto Pelayo Joint with Pedro Garc a S anchez and Gautam Webb* * =

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Computing the delta set of an affine semigroup: a status report

Christopher O’Neill

San Diego State University cdoneill@sdsu.edu Joint with Thomas Barron* and Roberto Pelayo Joint with Pedro Garc´ ıa S´ anchez and Gautam Webb* * = undergraduate student

Feb 10, 2020

Christopher O’Neill (SDSU) Computing delta sets of affine semigroups Feb 10, 2020 1 / 27

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Affine semigroups

Definition

A numerical semigroup S ⊂ Z≥0: closed under addition.

Christopher O’Neill (SDSU) Computing delta sets of affine semigroups Feb 10, 2020 2 / 27

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Affine semigroups

Definition

A numerical semigroup S ⊂ Z≥0: closed under addition.

Example

McN = 6, 9, 20 = {0, 6, 9, 12, 15, 18, 20, . . .}.

Christopher O’Neill (SDSU) Computing delta sets of affine semigroups Feb 10, 2020 2 / 27

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Affine semigroups

Definition

A numerical semigroup S ⊂ Z≥0: closed under addition.

Example

McN = 6, 9, 20 = {0, 6, 9, 12, 15, 18, 20, . . .}. “McNugget Semigroup”

Christopher O’Neill (SDSU) Computing delta sets of affine semigroups Feb 10, 2020 2 / 27

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Affine semigroups

Definition

A numerical semigroup S ⊂ Z≥0: closed under addition.

Example

McN = 6, 9, 20 = {0, 6, 9, 12, 15, 18, 20, . . .}. “McNugget Semigroup” Factorizations: 60 =

Christopher O’Neill (SDSU) Computing delta sets of affine semigroups Feb 10, 2020 2 / 27

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Affine semigroups

Definition

A numerical semigroup S ⊂ Z≥0: closed under addition.

Example

McN = 6, 9, 20 = {0, 6, 9, 12, 15, 18, 20, . . .}. “McNugget Semigroup” Factorizations: 60 = 7(6) + 2(9)

Christopher O’Neill (SDSU) Computing delta sets of affine semigroups Feb 10, 2020 2 / 27

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Affine semigroups

Definition

A numerical semigroup S ⊂ Z≥0: closed under addition.

Example

McN = 6, 9, 20 = {0, 6, 9, 12, 15, 18, 20, . . .}. “McNugget Semigroup” Factorizations: 60 = 7(6) + 2(9) = 3(20)

Christopher O’Neill (SDSU) Computing delta sets of affine semigroups Feb 10, 2020 2 / 27

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Affine semigroups

Definition

A numerical semigroup S ⊂ Z≥0: closed under addition.

Example

McN = 6, 9, 20 = {0, 6, 9, 12, 15, 18, 20, . . .}. “McNugget Semigroup” Factorizations: 60 = 7(6) + 2(9) = 3(20)

  • (7, 2, 0)

(0, 0, 3)

Christopher O’Neill (SDSU) Computing delta sets of affine semigroups Feb 10, 2020 2 / 27

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Affine semigroups

Definition

An affine semigroup S ⊂ Zd

≥0: closed under vector addition.

Christopher O’Neill (SDSU) Computing delta sets of affine semigroups Feb 10, 2020 3 / 27

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Affine semigroups

Definition

An affine semigroup S ⊂ Zd

≥0: closed under vector addition.

Example

S = (1, 1), (1, 5), (2, 5), (3, 5), (5, 1), (5, 2), (5, 3) ⊂ Z2

≥0

Christopher O’Neill (SDSU) Computing delta sets of affine semigroups Feb 10, 2020 3 / 27

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Affine semigroups

Definition

An affine semigroup S ⊂ Zd

≥0: closed under vector addition.

Example

S = (1, 1), (1, 5), (2, 5), (3, 5), (5, 1), (5, 2), (5, 3) ⊂ Z2

≥0

Christopher O’Neill (SDSU) Computing delta sets of affine semigroups Feb 10, 2020 3 / 27

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Affine semigroups

Definition

An affine semigroup S ⊂ Zd

≥0: closed under vector addition.

Example

S = (1, 1), (1, 5), (2, 5), (3, 5), (5, 1), (5, 2), (5, 3) ⊂ Z2

≥0

Christopher O’Neill (SDSU) Computing delta sets of affine semigroups Feb 10, 2020 3 / 27

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Affine semigroups

Definition

An affine semigroup S ⊂ Zd

≥0: closed under vector addition.

Example

S = (1, 1), (1, 5), (2, 5), (3, 5), (5, 1), (5, 2), (5, 3) ⊂ Z2

≥0

Z((6, 6)) =

  • (6, 0, 0, 0, 0, 0, 0),

(0, 1, 0, 0, 1, 0, 0)

  • Christopher O’Neill (SDSU)

Computing delta sets of affine semigroups Feb 10, 2020 3 / 27

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Affine semigroups

Definition

An affine semigroup S ⊂ Zd

≥0: closed under vector addition.

Example

S = (1, 1), (1, 5), (2, 5), (3, 5), (5, 1), (5, 2), (5, 3) ⊂ Z2

≥0

Z((6, 6)) =

  • (6, 0, 0, 0, 0, 0, 0),

(0, 1, 0, 0, 1, 0, 0)

  • (6, 6)

= 6(1, 1) = (1, 5) + (5, 1)

Christopher O’Neill (SDSU) Computing delta sets of affine semigroups Feb 10, 2020 3 / 27

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Affine semigroups

Definition

An affine semigroup S ⊂ Zd

≥0: closed under vector addition.

Example

S = (1, 1), (1, 5), (2, 5), (3, 5), (5, 1), (5, 2), (5, 3) ⊂ Z2

≥0

Z((6, 6)) =

  • (6, 0, 0, 0, 0, 0, 0),

(0, 1, 0, 0, 1, 0, 0)

  • (6, 6)

= 6(1, 1) = (1, 5) + (5, 1) S numerical semigroup

  • d = 1

Christopher O’Neill (SDSU) Computing delta sets of affine semigroups Feb 10, 2020 3 / 27

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Affine semigroups

Fix an affine semigroup S = n1, . . . , nk ⊂ Zd

≥0.

Z(n) =

  • a ∈ Zk

≥0 : n = a1n1 + · · · + aknk

  • is the set of factorizations of n ∈ S.

Christopher O’Neill (SDSU) Computing delta sets of affine semigroups Feb 10, 2020 4 / 27

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Affine semigroups

Fix an affine semigroup S = n1, . . . , nk ⊂ Zd

≥0.

Z(n) =

  • a ∈ Zk

≥0 : n = a1n1 + · · · + aknk

  • is the set of factorizations of n ∈ S.

|a| = a1 + · · · + ak (length of a)

Christopher O’Neill (SDSU) Computing delta sets of affine semigroups Feb 10, 2020 4 / 27

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SLIDE 18

Affine semigroups

Fix an affine semigroup S = n1, . . . , nk ⊂ Zd

≥0.

Z(n) =

  • a ∈ Zk

≥0 : n = a1n1 + · · · + aknk

  • is the set of factorizations of n ∈ S.

|a| = a1 + · · · + ak (length of a)

Example

S = 6, 9, 20: Z(60) = {(10, 0, 0), (7, 2, 0), (4, 4, 0), (1, 6, 0), (0, 0, 3)}

Christopher O’Neill (SDSU) Computing delta sets of affine semigroups Feb 10, 2020 4 / 27

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Affine semigroups

Fix an affine semigroup S = n1, . . . , nk ⊂ Zd

≥0.

Z(n) =

  • a ∈ Zk

≥0 : n = a1n1 + · · · + aknk

  • is the set of factorizations of n ∈ S.

|a| = a1 + · · · + ak (length of a)

Example

S = 6, 9, 20: Z(60) = {(10, 0, 0), (7, 2, 0), (4, 4, 0), (1, 6, 0), (0, 0, 3)} Possible factorization lengths for n = 60: 3, 7, 8, 9, 10.

Christopher O’Neill (SDSU) Computing delta sets of affine semigroups Feb 10, 2020 4 / 27

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Affine semigroups

Fix an affine semigroup S = n1, . . . , nk ⊂ Zd

≥0.

Z(n) =

  • a ∈ Zk

≥0 : n = a1n1 + · · · + aknk

  • is the set of factorizations of n ∈ S.

|a| = a1 + · · · + ak (length of a)

Example

S = 6, 9, 20: Z(60) = {(10, 0, 0), (7, 2, 0), (4, 4, 0), (1, 6, 0), (0, 0, 3)} Possible factorization lengths for n = 60: 3, 7, 8, 9, 10. Z(1000001) =

Christopher O’Neill (SDSU) Computing delta sets of affine semigroups Feb 10, 2020 4 / 27

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Affine semigroups

Fix an affine semigroup S = n1, . . . , nk ⊂ Zd

≥0.

Z(n) =

  • a ∈ Zk

≥0 : n = a1n1 + · · · + aknk

  • is the set of factorizations of n ∈ S.

|a| = a1 + · · · + ak (length of a)

Example

S = 6, 9, 20: Z(60) = {(10, 0, 0), (7, 2, 0), (4, 4, 0), (1, 6, 0), (0, 0, 3)} Possible factorization lengths for n = 60: 3, 7, 8, 9, 10. Z(1000001) = {

  • shortest

, . . . ,

  • longest

}

Christopher O’Neill (SDSU) Computing delta sets of affine semigroups Feb 10, 2020 4 / 27

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Affine semigroups

Fix an affine semigroup S = n1, . . . , nk ⊂ Zd

≥0.

Z(n) =

  • a ∈ Zk

≥0 : n = a1n1 + · · · + aknk

  • is the set of factorizations of n ∈ S.

|a| = a1 + · · · + ak (length of a)

Example

S = 6, 9, 20: Z(60) = {(10, 0, 0), (7, 2, 0), (4, 4, 0), (1, 6, 0), (0, 0, 3)} Possible factorization lengths for n = 60: 3, 7, 8, 9, 10. Z(1000001) = { (2, 1, 49999)

  • shortest

, . . . ,

  • longest

}

Christopher O’Neill (SDSU) Computing delta sets of affine semigroups Feb 10, 2020 4 / 27

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Affine semigroups

Fix an affine semigroup S = n1, . . . , nk ⊂ Zd

≥0.

Z(n) =

  • a ∈ Zk

≥0 : n = a1n1 + · · · + aknk

  • is the set of factorizations of n ∈ S.

|a| = a1 + · · · + ak (length of a)

Example

S = 6, 9, 20: Z(60) = {(10, 0, 0), (7, 2, 0), (4, 4, 0), (1, 6, 0), (0, 0, 3)} Possible factorization lengths for n = 60: 3, 7, 8, 9, 10. Z(1000001) = { (2, 1, 49999)

  • shortest

, . . . , (166662, 1, 1)

  • longest

}

Christopher O’Neill (SDSU) Computing delta sets of affine semigroups Feb 10, 2020 4 / 27

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Affine semigroups (a geometric viewpoint)

Fix an affine semigroup S = n1, . . . , nk ⊂ Zd

≥0.

Z(n) =

  • a ∈ Zk

≥0 : n = a1n1 + · · · + aknk

  • Christopher O’Neill (SDSU)

Computing delta sets of affine semigroups Feb 10, 2020 5 / 27

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Affine semigroups (a geometric viewpoint)

Fix an affine semigroup S = n1, . . . , nk ⊂ Zd

≥0.

Z(n) =

  • a ∈ Zk

≥0 : n = a1n1 + · · · + aknk

  • A geometric viewpoint: non-negative integer solutions to linear equations.

Christopher O’Neill (SDSU) Computing delta sets of affine semigroups Feb 10, 2020 5 / 27

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Affine semigroups (a geometric viewpoint)

Fix an affine semigroup S = n1, . . . , nk ⊂ Zd

≥0.

Z(n) =

  • a ∈ Zk

≥0 : n = a1n1 + · · · + aknk

  • A geometric viewpoint: non-negative integer solutions to linear equations.

Example: S = 6, 9, 20, n = 60.

Christopher O’Neill (SDSU) Computing delta sets of affine semigroups Feb 10, 2020 5 / 27

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SLIDE 27

Affine semigroups (a geometric viewpoint)

Fix an affine semigroup S = n1, . . . , nk ⊂ Zd

≥0.

Z(n) =

  • a ∈ Zk

≥0 : n = a1n1 + · · · + aknk

  • A geometric viewpoint: non-negative integer solutions to linear equations.

Example: S = 6, 9, 20, n = 60. Z(60) =

  • a ∈ Zk

≥0 : 60 = 6a1 + 9a2 + 20a3

  • Christopher O’Neill (SDSU)

Computing delta sets of affine semigroups Feb 10, 2020 5 / 27

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Affine semigroups (a geometric viewpoint)

Fix an affine semigroup S = n1, . . . , nk ⊂ Zd

≥0.

Z(n) =

  • a ∈ Zk

≥0 : n = a1n1 + · · · + aknk

  • A geometric viewpoint: non-negative integer solutions to linear equations.

Example: S = 6, 9, 20, n = 60. Z(60) =

  • a ∈ Zk

≥0 : 60 = 6a1 + 9a2 + 20a3

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T j 5 z D H / g f P 4 A W V S N w A = = < / l a t e x i t >

Christopher O’Neill (SDSU) Computing delta sets of affine semigroups Feb 10, 2020 5 / 27

slide-29
SLIDE 29

Delta set

Fix an affine semigroup S = n1, . . . , nk ⊂ Zd

≥0.

Z(n) =

  • a ∈ Zk

≥0 : n = a1n1 + · · · + aknk

  • Christopher O’Neill (SDSU)

Computing delta sets of affine semigroups Feb 10, 2020 6 / 27

slide-30
SLIDE 30

Delta set

Fix an affine semigroup S = n1, . . . , nk ⊂ Zd

≥0.

Z(n) =

  • a ∈ Zk

≥0 : n = a1n1 + · · · + aknk

  • a = (a1, . . . , ak) ∈ Zk

≥0

  • |a| = a1 + · · · + ak

Christopher O’Neill (SDSU) Computing delta sets of affine semigroups Feb 10, 2020 6 / 27

slide-31
SLIDE 31

Delta set

Fix an affine semigroup S = n1, . . . , nk ⊂ Zd

≥0.

Z(n) =

  • a ∈ Zk

≥0 : n = a1n1 + · · · + aknk

  • a = (a1, . . . , ak) ∈ Zk

≥0

  • |a| = a1 + · · · + ak

L(n) = {|a| : a ∈ Z(n)} = {ℓ1 < · · · < ℓr}

Christopher O’Neill (SDSU) Computing delta sets of affine semigroups Feb 10, 2020 6 / 27

slide-32
SLIDE 32

Delta set

Fix an affine semigroup S = n1, . . . , nk ⊂ Zd

≥0.

Z(n) =

  • a ∈ Zk

≥0 : n = a1n1 + · · · + aknk

  • a = (a1, . . . , ak) ∈ Zk

≥0

  • |a| = a1 + · · · + ak

L(n) = {|a| : a ∈ Z(n)} = {ℓ1 < · · · < ℓr}

Definition

The delta set of n ∈ S is ∆(n) = {ℓ2 − ℓ1, . . . , ℓr − ℓr−1}

Christopher O’Neill (SDSU) Computing delta sets of affine semigroups Feb 10, 2020 6 / 27

slide-33
SLIDE 33

Delta set

Fix an affine semigroup S = n1, . . . , nk ⊂ Zd

≥0.

Z(n) =

  • a ∈ Zk

≥0 : n = a1n1 + · · · + aknk

  • a = (a1, . . . , ak) ∈ Zk

≥0

  • |a| = a1 + · · · + ak

L(n) = {|a| : a ∈ Z(n)} = {ℓ1 < · · · < ℓr}

Definition

The delta set of n ∈ S is ∆(n) = {ℓ2 − ℓ1, . . . , ℓr − ℓr−1}

Example

S = 6, 9, 20: Z(60) = {(10, 0, 0), (7, 2, 0), (4, 4, 0), (1, 6, 0), (0, 0, 3)}

Christopher O’Neill (SDSU) Computing delta sets of affine semigroups Feb 10, 2020 6 / 27

slide-34
SLIDE 34

Delta set

Fix an affine semigroup S = n1, . . . , nk ⊂ Zd

≥0.

Z(n) =

  • a ∈ Zk

≥0 : n = a1n1 + · · · + aknk

  • a = (a1, . . . , ak) ∈ Zk

≥0

  • |a| = a1 + · · · + ak

L(n) = {|a| : a ∈ Z(n)} = {ℓ1 < · · · < ℓr}

Definition

The delta set of n ∈ S is ∆(n) = {ℓ2 − ℓ1, . . . , ℓr − ℓr−1}

Example

S = 6, 9, 20: Z(60) = {(10, 0, 0), (7, 2, 0), (4, 4, 0), (1, 6, 0), (0, 0, 3)} L(60) = {3, 7, 8, 9, 10} ∆(60) = {1, 4}

Christopher O’Neill (SDSU) Computing delta sets of affine semigroups Feb 10, 2020 6 / 27

slide-35
SLIDE 35

Delta set

Fix an affine semigroup S = n1, . . . , nk ⊂ Zd

≥0.

Z(n) =

  • a ∈ Zk

≥0 : n = a1n1 + · · · + aknk

  • a = (a1, . . . , ak) ∈ Zk

≥0

  • |a| = a1 + · · · + ak

L(n) = {|a| : a ∈ Z(n)} = {ℓ1 < · · · < ℓr}

Definition

The delta set of n ∈ S is ∆(n) = {ℓ2 − ℓ1, . . . , ℓr − ℓr−1}

Example

S = 6, 9, 20: Z(60) = {(10, 0, 0), (7, 2, 0), (4, 4, 0), (1, 6, 0), (0, 0, 3)} L(60) = {3, 7, 8, 9, 10} ∆(60) = {1, 4} L(142) = {10, 11, 12, 14, 15, 16, 17, 18, 19} ∆(142) = {1, 2}

Christopher O’Neill (SDSU) Computing delta sets of affine semigroups Feb 10, 2020 6 / 27

slide-36
SLIDE 36

Delta set

Fix an affine semigroup S = n1, . . . , nk ⊂ Zd

≥0.

Z(n) =

  • a ∈ Zk

≥0 : n = a1n1 + · · · + aknk

  • a = (a1, . . . , ak) ∈ Zk

≥0

  • |a| = a1 + · · · + ak

L(n) = {ℓ1 < · · · < ℓr} ∆(n) = {ℓi − ℓi−1}

Christopher O’Neill (SDSU) Computing delta sets of affine semigroups Feb 10, 2020 7 / 27

slide-37
SLIDE 37

Delta set

Fix an affine semigroup S = n1, . . . , nk ⊂ Zd

≥0.

Z(n) =

  • a ∈ Zk

≥0 : n = a1n1 + · · · + aknk

  • a = (a1, . . . , ak) ∈ Zk

≥0

  • |a| = a1 + · · · + ak

L(n) = {ℓ1 < · · · < ℓr} ∆(n) = {ℓi − ℓi−1}

Facts for large n ∈ S

min ∆(n) is as small as possible for S

Christopher O’Neill (SDSU) Computing delta sets of affine semigroups Feb 10, 2020 7 / 27

slide-38
SLIDE 38

Delta set

Fix an affine semigroup S = n1, . . . , nk ⊂ Zd

≥0.

Z(n) =

  • a ∈ Zk

≥0 : n = a1n1 + · · · + aknk

  • a = (a1, . . . , ak) ∈ Zk

≥0

  • |a| = a1 + · · · + ak

L(n) = {ℓ1 < · · · < ℓr} ∆(n) = {ℓi − ℓi−1}

Facts for large n ∈ S

min ∆(n) is as small as possible for S S = 3, 5, 7: L(110) = {16, 18, . . . , 34, 36}, ∆(110) = {2}

Christopher O’Neill (SDSU) Computing delta sets of affine semigroups Feb 10, 2020 7 / 27

slide-39
SLIDE 39

Delta set

Fix an affine semigroup S = n1, . . . , nk ⊂ Zd

≥0.

Z(n) =

  • a ∈ Zk

≥0 : n = a1n1 + · · · + aknk

  • a = (a1, . . . , ak) ∈ Zk

≥0

  • |a| = a1 + · · · + ak

L(n) = {ℓ1 < · · · < ℓr} ∆(n) = {ℓi − ℓi−1}

Facts for large n ∈ S

min ∆(n) is as small as possible for S S = 3, 5, 7: L(110) = {16, 18, . . . , 34, 36}, ∆(110) = {2} L(n) is an arithmetic sequence with a few values removed near the ends

Christopher O’Neill (SDSU) Computing delta sets of affine semigroups Feb 10, 2020 7 / 27

slide-40
SLIDE 40

Delta set

Fix an affine semigroup S = n1, . . . , nk ⊂ Zd

≥0.

Z(n) =

  • a ∈ Zk

≥0 : n = a1n1 + · · · + aknk

  • a = (a1, . . . , ak) ∈ Zk

≥0

  • |a| = a1 + · · · + ak

L(n) = {ℓ1 < · · · < ℓr} ∆(n) = {ℓi − ℓi−1}

Facts for large n ∈ S

min ∆(n) is as small as possible for S S = 3, 5, 7: L(110) = {16, 18, . . . , 34, 36}, ∆(110) = {2} L(n) is an arithmetic sequence with a few values removed near the ends S = 42, 86, 245, 285, 365, 463: L(3023) = {7, 9, 11, 12, . . . , 46, 47, 58, 62, 64}, ∆(3023) = {1, 2, 4, 9}

Christopher O’Neill (SDSU) Computing delta sets of affine semigroups Feb 10, 2020 7 / 27

slide-41
SLIDE 41

Delta set (a geometric viewpoint)

Fix an affine semigroup S = n1, . . . , nk ⊂ Zd

≥0.

Z(n) =

  • a ∈ Zk

≥0 : n = a1n1 + · · · + aknk

  • a = (a1, . . . , ak) ∈ Zk

≥0

  • |a| = a1 + · · · + ak

L(n) = {ℓ1 < · · · < ℓr} ∆(n) = {ℓi − ℓi−1}

Christopher O’Neill (SDSU) Computing delta sets of affine semigroups Feb 10, 2020 8 / 27

slide-42
SLIDE 42

Delta set (a geometric viewpoint)

Fix an affine semigroup S = n1, . . . , nk ⊂ Zd

≥0.

Z(n) =

  • a ∈ Zk

≥0 : n = a1n1 + · · · + aknk

  • a = (a1, . . . , ak) ∈ Zk

≥0

  • |a| = a1 + · · · + ak

L(n) = {ℓ1 < · · · < ℓr} ∆(n) = {ℓi − ℓi−1} A geometric viewpoint: lattice width

Christopher O’Neill (SDSU) Computing delta sets of affine semigroups Feb 10, 2020 8 / 27

slide-43
SLIDE 43

Delta set (a geometric viewpoint)

Fix an affine semigroup S = n1, . . . , nk ⊂ Zd

≥0.

Z(n) =

  • a ∈ Zk

≥0 : n = a1n1 + · · · + aknk

  • a = (a1, . . . , ak) ∈ Zk

≥0

  • |a| = a1 + · · · + ak

L(n) = {ℓ1 < · · · < ℓr} ∆(n) = {ℓi − ℓi−1} A geometric viewpoint: lattice width

Christopher O’Neill (SDSU) Computing delta sets of affine semigroups Feb 10, 2020 8 / 27

slide-44
SLIDE 44

Delta set (a geometric viewpoint)

Fix an affine semigroup S = n1, . . . , nk ⊂ Zd

≥0.

Z(n) =

  • a ∈ Zk

≥0 : n = a1n1 + · · · + aknk

  • a = (a1, . . . , ak) ∈ Zk

≥0

  • |a| = a1 + · · · + ak

L(n) = {ℓ1 < · · · < ℓr} ∆(n) = {ℓi − ℓi−1} A geometric viewpoint: lattice width |a| = a1 (the ℓ1-norm)

Christopher O’Neill (SDSU) Computing delta sets of affine semigroups Feb 10, 2020 8 / 27

slide-45
SLIDE 45

Delta set (a geometric viewpoint)

Fix an affine semigroup S = n1, . . . , nk ⊂ Zd

≥0.

Z(n) =

  • a ∈ Zk

≥0 : n = a1n1 + · · · + aknk

  • a = (a1, . . . , ak) ∈ Zk

≥0

  • |a| = a1 + · · · + ak

L(n) = {ℓ1 < · · · < ℓr} ∆(n) = {ℓi − ℓi−1} A geometric viewpoint: lattice width |a| = a1 (the ℓ1-norm) min ∆(n): min ℓ1 width

Christopher O’Neill (SDSU) Computing delta sets of affine semigroups Feb 10, 2020 8 / 27

slide-46
SLIDE 46

Delta set (a geometric viewpoint)

Fix an affine semigroup S = n1, . . . , nk ⊂ Zd

≥0.

Z(n) =

  • a ∈ Zk

≥0 : n = a1n1 + · · · + aknk

  • a = (a1, . . . , ak) ∈ Zk

≥0

  • |a| = a1 + · · · + ak

L(n) = {ℓ1 < · · · < ℓr} ∆(n) = {ℓi − ℓi−1} A geometric viewpoint: lattice width |a| = a1 (the ℓ1-norm) min ∆(n): min ℓ1 width extremal lengths near vertices

Christopher O’Neill (SDSU) Computing delta sets of affine semigroups Feb 10, 2020 8 / 27

slide-47
SLIDE 47

Delta set of a semigroup

Fix an affine semigroup S = n1, . . . , nk ⊂ Zd

≥0.

Z(n) =

  • a ∈ Zk

≥0 : n = a1n1 + · · · + aknk

  • a = (a1, . . . , ak) ∈ Zk

≥0

  • |a| = a1 + · · · + ak

L(n) = {ℓ1 < · · · < ℓr} ∆(n) = {ℓi − ℓi−1}

Christopher O’Neill (SDSU) Computing delta sets of affine semigroups Feb 10, 2020 9 / 27

slide-48
SLIDE 48

Delta set of a semigroup

Fix an affine semigroup S = n1, . . . , nk ⊂ Zd

≥0.

Z(n) =

  • a ∈ Zk

≥0 : n = a1n1 + · · · + aknk

  • a = (a1, . . . , ak) ∈ Zk

≥0

  • |a| = a1 + · · · + ak

L(n) = {ℓ1 < · · · < ℓr} ∆(n) = {ℓi − ℓi−1}

Definition

The delta set of S is the union ∆(S) =

n∈S ∆(n)

Christopher O’Neill (SDSU) Computing delta sets of affine semigroups Feb 10, 2020 9 / 27

slide-49
SLIDE 49

Delta set of a semigroup

Fix an affine semigroup S = n1, . . . , nk ⊂ Zd

≥0.

Z(n) =

  • a ∈ Zk

≥0 : n = a1n1 + · · · + aknk

  • a = (a1, . . . , ak) ∈ Zk

≥0

  • |a| = a1 + · · · + ak

L(n) = {ℓ1 < · · · < ℓr} ∆(n) = {ℓi − ℓi−1}

Definition

The delta set of S is the union ∆(S) =

n∈S ∆(n)

Goal

Given n1, . . . , nk as input, compute ∆(S).

Christopher O’Neill (SDSU) Computing delta sets of affine semigroups Feb 10, 2020 9 / 27

slide-50
SLIDE 50

Delta set of a semigroup

Fix an affine semigroup S = n1, . . . , nk ⊂ Zd

≥0.

Z(n) =

  • a ∈ Zk

≥0 : n = a1n1 + · · · + aknk

  • a = (a1, . . . , ak) ∈ Zk

≥0

  • |a| = a1 + · · · + ak

L(n) = {ℓ1 < · · · < ℓr} ∆(n) = {ℓi − ℓi−1}

Definition

The delta set of S is the union ∆(S) =

n∈S ∆(n)

Goal

Given n1, . . . , nk as input, compute ∆(S). The primary difficulty: this is an infinite union!

Christopher O’Neill (SDSU) Computing delta sets of affine semigroups Feb 10, 2020 9 / 27

slide-51
SLIDE 51

Delta set of a semigroup

Fix an affine semigroup S = n1, . . . , nk ⊂ Zd

≥0.

Z(n) =

  • a ∈ Zk

≥0 : n = a1n1 + · · · + aknk

  • a = (a1, . . . , ak) ∈ Zk

≥0

  • |a| = a1 + · · · + ak

L(n) = {ℓ1 < · · · < ℓr} ∆(n) = {ℓi − ℓi−1}

Definition

The delta set of S is the union ∆(S) =

n∈S ∆(n)

Goal

Given n1, . . . , nk as input, compute ∆(S). The primary difficulty: this is an infinite union!

Example

S = 17, 33, 53, 71, ∆(S) = {2, 4, 6}.

Christopher O’Neill (SDSU) Computing delta sets of affine semigroups Feb 10, 2020 9 / 27

slide-52
SLIDE 52

Delta set of a semigroup

Fix an affine semigroup S = n1, . . . , nk ⊂ Zd

≥0.

Z(n) =

  • a ∈ Zk

≥0 : n = a1n1 + · · · + aknk

  • a = (a1, . . . , ak) ∈ Zk

≥0

  • |a| = a1 + · · · + ak

L(n) = {ℓ1 < · · · < ℓr} ∆(n) = {ℓi − ℓi−1}

Definition

The delta set of S is the union ∆(S) =

n∈S ∆(n)

Goal

Given n1, . . . , nk as input, compute ∆(S). The primary difficulty: this is an infinite union!

Example

S = 17, 33, 53, 71, ∆(S) = {2, 4, 6}. 6 ∈ ∆(266), ∆(283), ∆(300).

Christopher O’Neill (SDSU) Computing delta sets of affine semigroups Feb 10, 2020 9 / 27

slide-53
SLIDE 53

The first breakthrough: numerical semigroups

Fix a numerical semigroup S = n1, . . . , nk ⊂ Z≥0.

20 40 60 80 100 120 140 1 2 3 4 Christopher O’Neill (SDSU) Computing delta sets of affine semigroups Feb 10, 2020 10 / 27

slide-54
SLIDE 54

The first breakthrough: numerical semigroups

Fix a numerical semigroup S = n1, . . . , nk ⊂ Z≥0.

Theorem (Chapman–Hoyer–Kaplan, 2009)

For n ≥ 2kn2n2

k, ∆(n) = ∆(n + n1nk).

20 40 60 80 100 120 140 1 2 3 4 Christopher O’Neill (SDSU) Computing delta sets of affine semigroups Feb 10, 2020 10 / 27

slide-55
SLIDE 55

The first breakthrough: numerical semigroups

Fix a numerical semigroup S = n1, . . . , nk ⊂ Z≥0.

Theorem (Chapman–Hoyer–Kaplan, 2009)

For n ≥ 2kn2n2

k, ∆(n) = ∆(n + n1nk).

Example: S = 6, 9, 20: 2kn2n2

k = 21600

20 40 60 80 100 120 140 1 2 3 4

(n, d)

  • d ∈ ∆(n)

Christopher O’Neill (SDSU) Computing delta sets of affine semigroups Feb 10, 2020 10 / 27

slide-56
SLIDE 56

The first breakthrough: numerical semigroups

Fix a numerical semigroup S = n1, . . . , nk ⊂ Z≥0.

Theorem (Chapman–Hoyer–Kaplan, 2009)

For n ≥ 2kn2n2

k, ∆(n) = ∆(n + n1nk).

Christopher O’Neill (SDSU) Computing delta sets of affine semigroups Feb 10, 2020 11 / 27

slide-57
SLIDE 57

The first breakthrough: numerical semigroups

Fix a numerical semigroup S = n1, . . . , nk ⊂ Z≥0.

Theorem (Chapman–Hoyer–Kaplan, 2009)

For n ≥ 2kn2n2

k, ∆(n) = ∆(n + n1nk).

For n ∈ S with 0 ≤ n ≤ 2kn2n2

k + n1nk,

Christopher O’Neill (SDSU) Computing delta sets of affine semigroups Feb 10, 2020 11 / 27

slide-58
SLIDE 58

The first breakthrough: numerical semigroups

Fix a numerical semigroup S = n1, . . . , nk ⊂ Z≥0.

Theorem (Chapman–Hoyer–Kaplan, 2009)

For n ≥ 2kn2n2

k, ∆(n) = ∆(n + n1nk).

For n ∈ S with 0 ≤ n ≤ 2kn2n2

k + n1nk,

compute: Z(n) = {a ∈ Zk

≥0 : n = a1n1 + · · · + aknk}

Christopher O’Neill (SDSU) Computing delta sets of affine semigroups Feb 10, 2020 11 / 27

slide-59
SLIDE 59

The first breakthrough: numerical semigroups

Fix a numerical semigroup S = n1, . . . , nk ⊂ Z≥0.

Theorem (Chapman–Hoyer–Kaplan, 2009)

For n ≥ 2kn2n2

k, ∆(n) = ∆(n + n1nk).

For n ∈ S with 0 ≤ n ≤ 2kn2n2

k + n1nk,

compute: Z(n) = {a ∈ Zk

≥0 : n = a1n1 + · · · + aknk}

Z(n) L(n) = {a1 + · · · + ak : a ∈ Z(n)}

Christopher O’Neill (SDSU) Computing delta sets of affine semigroups Feb 10, 2020 11 / 27

slide-60
SLIDE 60

The first breakthrough: numerical semigroups

Fix a numerical semigroup S = n1, . . . , nk ⊂ Z≥0.

Theorem (Chapman–Hoyer–Kaplan, 2009)

For n ≥ 2kn2n2

k, ∆(n) = ∆(n + n1nk).

For n ∈ S with 0 ≤ n ≤ 2kn2n2

k + n1nk,

compute: Z(n) = {a ∈ Zk

≥0 : n = a1n1 + · · · + aknk}

Z(n) L(n) = {a1 + · · · + ak : a ∈ Z(n)} L(n) = {ℓ1 < . . . < ℓr} ∆(n) = {ℓi − ℓi−1}

Christopher O’Neill (SDSU) Computing delta sets of affine semigroups Feb 10, 2020 11 / 27

slide-61
SLIDE 61

The first breakthrough: numerical semigroups

Fix a numerical semigroup S = n1, . . . , nk ⊂ Z≥0.

Theorem (Chapman–Hoyer–Kaplan, 2009)

For n ≥ 2kn2n2

k, ∆(n) = ∆(n + n1nk).

For n ∈ S with 0 ≤ n ≤ 2kn2n2

k + n1nk,

compute: Z(n) = {a ∈ Zk

≥0 : n = a1n1 + · · · + aknk}

Z(n) L(n) = {a1 + · · · + ak : a ∈ Z(n)} L(n) = {ℓ1 < . . . < ℓr} ∆(n) = {ℓi − ℓi−1} Compute ∆(S) =

n ∆(n).

Christopher O’Neill (SDSU) Computing delta sets of affine semigroups Feb 10, 2020 11 / 27

slide-62
SLIDE 62

The first breakthrough: numerical semigroups

Fix a numerical semigroup S = n1, . . . , nk ⊂ Z≥0.

Theorem (Chapman–Hoyer–Kaplan, 2009)

For n ≥ 2kn2n2

k, ∆(n) = ∆(n + n1nk).

For n ∈ S with 0 ≤ n ≤ 2kn2n2

k + n1nk,

compute: → Z(n) = {a ∈ Zk

≥0 : n = a1n1 + · · · + aknk} ←

Z(n) L(n) = {a1 + · · · + ak : a ∈ Z(n)} L(n) = {ℓ1 < . . . < ℓr} ∆(n) = {ℓi − ℓi−1} Compute ∆(S) =

n ∆(n).

Christopher O’Neill (SDSU) Computing delta sets of affine semigroups Feb 10, 2020 11 / 27

slide-63
SLIDE 63

The first breakthrough: numerical semigroups

Fix a numerical semigroup S = n1, . . . , nk ⊂ Z≥0.

Theorem (Chapman–Hoyer–Kaplan, 2009)

For n ≥ 2kn2n2

k, ∆(n) = ∆(n + n1nk).

For n ∈ S with 0 ≤ n ≤ 2kn2n2

k + n1nk,

compute: → Z(n) = {a ∈ Zk

≥0 : n = a1n1 + · · · + aknk} ←

Z(n) L(n) = {a1 + · · · + ak : a ∈ Z(n)} L(n) = {ℓ1 < . . . < ℓr} ∆(n) = {ℓi − ℓi−1} Compute ∆(S) =

n ∆(n).

|Z(n)| ≈ nk−1

Christopher O’Neill (SDSU) Computing delta sets of affine semigroups Feb 10, 2020 11 / 27

slide-64
SLIDE 64

An improvement: the first usable algorithm

Fix a numerical semigroup S = n1, . . . , nk ⊂ Z≥0.

Theorem (Chapman–Hoyer–Kaplan, 2009)

For n ≥ 2kn2n2

k, ∆(n) = ∆(n + n1nk).

Christopher O’Neill (SDSU) Computing delta sets of affine semigroups Feb 10, 2020 12 / 27

slide-65
SLIDE 65

An improvement: the first usable algorithm

Fix a numerical semigroup S = n1, . . . , nk ⊂ Z≥0.

Theorem (Garc´ ıa-Garc´ ıa–Moreno-Fr´ ıas–Vigneron-Tenorio, 2014)

For n ≥ NS, ∆(n) = ∆(n + lcm(n1, nk)).

Christopher O’Neill (SDSU) Computing delta sets of affine semigroups Feb 10, 2020 12 / 27

slide-66
SLIDE 66

An improvement: the first usable algorithm

Fix a numerical semigroup S = n1, . . . , nk ⊂ Z≥0.

Theorem (Garc´ ıa-Garc´ ıa–Moreno-Fr´ ıas–Vigneron-Tenorio, 2014)

For n ≥ NS, ∆(n) = ∆(n + lcm(n1, nk)).

Christopher O’Neill (SDSU) Computing delta sets of affine semigroups Feb 10, 2020 12 / 27

slide-67
SLIDE 67

An improvement: the first usable algorithm

Fix a numerical semigroup S = n1, . . . , nk ⊂ Z≥0.

Theorem (Garc´ ıa-Garc´ ıa–Moreno-Fr´ ıas–Vigneron-Tenorio, 2014)

For n ≥ NS, ∆(n) = ∆(n + lcm(n1, nk)). S = 6, 9, 20: 2kn2n2

k = 21600, NS = 144, actual = 91

Christopher O’Neill (SDSU) Computing delta sets of affine semigroups Feb 10, 2020 12 / 27

slide-68
SLIDE 68

An improvement: the first usable algorithm

Fix a numerical semigroup S = n1, . . . , nk ⊂ Z≥0.

Theorem (Garc´ ıa-Garc´ ıa–Moreno-Fr´ ıas–Vigneron-Tenorio, 2014)

For n ≥ NS, ∆(n) = ∆(n + lcm(n1, nk)). S = 6, 9, 20: 2kn2n2

k = 21600, NS = 144, actual = 91

S 2kn2n2

k

NS ∆(S) Runtime 7, 15, 17, 18, 20 60000 1935 {1, 2, 3} 1m 28s 11, 53, 73, 87 3209256 14381 {2, 4, 6, 8, 10, 22} 0m 49s 31, 73, 77, 87, 91 6045130 31364 {2, 4, 6} 400m 12s 100, 121, 142, 163, 284 9 · 107 24850 ——— ——— 1001, 1211, 1421, 1631, 2841 6 · 1015 2063141 ——— ———

Christopher O’Neill (SDSU) Computing delta sets of affine semigroups Feb 10, 2020 12 / 27

slide-69
SLIDE 69

An improvement: the first usable algorithm

Fix a numerical semigroup S = n1, . . . , nk ⊂ Z≥0.

Theorem (Garc´ ıa-Garc´ ıa–Moreno-Fr´ ıas–Vigneron-Tenorio, 2014)

For n ≥ NS, ∆(n) = ∆(n + lcm(n1, nk)). S = 6, 9, 20: 2kn2n2

k = 21600, NS = 144, actual = 91

S 2kn2n2

k

NS ∆(S) Runtime 7, 15, 17, 18, 20 60000 1935 {1, 2, 3} 1m 28s 11, 53, 73, 87 3209256 14381 {2, 4, 6, 8, 10, 22} 0m 49s 31, 73, 77, 87, 91 6045130 31364 {2, 4, 6} 400m 12s 100, 121, 142, 163, 284 9 · 107 24850 ——— ——— 1001, 1211, 1421, 1631, 2841 6 · 1015 2063141 ——— ———

GAP Numerical Semigroups Package, available at http://www.gap-system.org/Packages/numericalsgps.html.

Christopher O’Neill (SDSU) Computing delta sets of affine semigroups Feb 10, 2020 12 / 27

slide-70
SLIDE 70

An improvement: the first usable algorithm

Fix a numerical semigroup S = n1, . . . , nk ⊂ Z≥0.

Theorem (Garc´ ıa-Garc´ ıa–Moreno-Fr´ ıas–Vigneron-Tenorio, 2014)

For n ≥ NS, ∆(n) = ∆(n + lcm(n1, nk)).

Christopher O’Neill (SDSU) Computing delta sets of affine semigroups Feb 10, 2020 13 / 27

slide-71
SLIDE 71

An improvement: the first usable algorithm

Fix a numerical semigroup S = n1, . . . , nk ⊂ Z≥0.

Theorem (Garc´ ıa-Garc´ ıa–Moreno-Fr´ ıas–Vigneron-Tenorio, 2014)

For n ≥ NS, ∆(n) = ∆(n + lcm(n1, nk)). For n ∈ S with 0 ≤ n ≤ 2kn2n2

k + n1nk,

compute: Z(n) = {a ∈ Zk

≥0 : n = a1n1 + · · · + aknk}

Z(n) L(n) = {a1 + · · · + ak : a ∈ Z(n)} L(n) = {ℓ1 < . . . < ℓr} ∆(n) = {ℓi − ℓi−1} Compute ∆(S) =

n ∆(n).

Christopher O’Neill (SDSU) Computing delta sets of affine semigroups Feb 10, 2020 13 / 27

slide-72
SLIDE 72

An improvement: the first usable algorithm

Fix a numerical semigroup S = n1, . . . , nk ⊂ Z≥0.

Theorem (Garc´ ıa-Garc´ ıa–Moreno-Fr´ ıas–Vigneron-Tenorio, 2014)

For n ≥ NS, ∆(n) = ∆(n + lcm(n1, nk)). For n ∈ S with 0 ≤ n ≤ 2kn2n2

k + n1nk,

compute: → Z(n) = {a ∈ Zk

≥0 : n = a1n1 + · · · + aknk} ←

Z(n) L(n) = {a1 + · · · + ak : a ∈ Z(n)} L(n) = {ℓ1 < . . . < ℓr} ∆(n) = {ℓi − ℓi−1} Compute ∆(S) =

n ∆(n).

Christopher O’Neill (SDSU) Computing delta sets of affine semigroups Feb 10, 2020 13 / 27

slide-73
SLIDE 73

A faster solution: dynamic programming

Fix n ∈ S = n1, . . . , nk.

Christopher O’Neill (SDSU) Computing delta sets of affine semigroups Feb 10, 2020 14 / 27

slide-74
SLIDE 74

A faster solution: dynamic programming

Fix n ∈ S = n1, . . . , nk. For each i ≤ k,

Christopher O’Neill (SDSU) Computing delta sets of affine semigroups Feb 10, 2020 14 / 27

slide-75
SLIDE 75

A faster solution: dynamic programming

Fix n ∈ S = n1, . . . , nk. For each i ≤ k, φi : Z(n − ni) − → Z(n) a − → a + ei

Christopher O’Neill (SDSU) Computing delta sets of affine semigroups Feb 10, 2020 14 / 27

slide-76
SLIDE 76

A faster solution: dynamic programming

Fix n ∈ S = n1, . . . , nk. For each i ≤ k, φi : Z(n − ni) − → Z(n) a − → a + ei S = 6, 9, 20:

Christopher O’Neill (SDSU) Computing delta sets of affine semigroups Feb 10, 2020 14 / 27

slide-77
SLIDE 77

A faster solution: dynamic programming

Fix n ∈ S = n1, . . . , nk. For each i ≤ k, φi : Z(n − ni) − → Z(n) a − → a + ei S = 6, 9, 20:

< l a t e x i t s h a 1 _ b a s e 6 4 = " y A
  • H
e 6 O H C j 8 O R 3 h l J Q l R T C X I N 8 = " > A A A B 9 X i c b V D L T g I x F L 2 D L 8 Q X 6 t J N I z H B D Z k h G H V H d O M S E 3 l E G E m n d K C h 5 m H Q 2 Z 8 B 9 u X G i M W / / F n X 9 j G W a h 4 E m a n J x z b + 7 p 8 S L O l L b t b y u 3 s r q 2 v p H f L G x t 7 + z u F f c P W i q M J a F N E v J Q d j y s K G e C N j X T n H Y i S X H g c d r 2 x t c z v / 1 I p W K h u N O T i L
  • B
H g r m M 4 K 1 k R 5 6 A d Y j 5 a P 7 c v X s 8 r R f L N k V O w V a J k 5 G S p C h S 9 + 9 Q Y h i Q M q N O F Y q a 5 j R 9 p N s N S M c D
  • t
9 G J F I z G e E i 7 h g
  • c
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O j H K A P m h N E 9
  • l
K q / N x I c K D U J P D O Z p l z Z u J / X j f W /
  • W
b M B H F m g
  • y
P + T H H O k Q z S p A A y Y p X x i C C a S m a y I j L D E R J u i C q Y E Z / H L y 6 R V r T h 2 x b m t l e p X W R 1 5 O I J j K I M D 5 1 C H G 2 h A E w h I e I Z X e L O e r B f r 3 f q Y j + a s b O c Q / s D 6 / A G y C Z F Q < / l a t e x i t >

Christopher O’Neill (SDSU) Computing delta sets of affine semigroups Feb 10, 2020 14 / 27

slide-78
SLIDE 78

A faster solution: dynamic programming

Fix n ∈ S = n1, . . . , nk. For each i ≤ k, φi : Z(n − ni) − → Z(n) a − → a + ei S = 6, 9, 20:

< l a t e x i t s h a 1 _ b a s e 6 4 = " y A
  • H
e 6 O H C j 8 O R 3 h l J Q l R T C X I N 8 = " > A A A B 9 X i c b V D L T g I x F L 2 D L 8 Q X 6 t J N I z H B D Z k h G H V H d O M S E 3 l E G E m n d K C h 5 m H Q 2 Z 8 B 9 u X G i M W / / F n X 9 j G W a h 4 E m a n J x z b + 7 p 8 S L O l L b t b y u 3 s r q 2 v p H f L G x t 7 + z u F f c P W i q M J a F N E v J Q d j y s K G e C N j X T n H Y i S X H g c d r 2 x t c z v / 1 I p W K h u N O T i L
  • B
H g r m M 4 K 1 k R 5 6 A d Y j 5 a P 7 c v X s 8 r R f L N k V O w V a J k 5 G S p C h S 9 + 9 Q Y h i Q M q N O F Y q a 5 j R 9 p N s N S M c D
  • t
9 G J F I z G e E i 7 h g
  • c
U O U m a e
  • p
O j H K A P m h N E 9
  • l
K q / N x I c K D U J P D O Z p l z Z u J / X j f W /
  • W
b M B H F m g
  • y
P + T H H O k Q z S p A A y Y p X x i C C a S m a y I j L D E R J u i C q Y E Z / H L y 6 R V r T h 2 x b m t l e p X W R 1 5 O I J j K I M D 5 1 C H G 2 h A E w h I e I Z X e L O e r B f r 3 f q Y j + a s b O c Q / s D 6 / A G y C Z F Q < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " 8 a m w z C u D a a J M U F O d I D w i V j h s U = " > A A A B 9 X i c b V D L T g I x F L 2 D L 8 Q X 6 t J N I z H B D Z k B j S 6 J b l x i I
  • 8
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  • y
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  • c
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  • S
U T f A Q 8 F 8 R r A 2 k M v w H q k f H R f r p 7 X T v v F k l 2 x U 6 B l 4 m S k B B k a / e J X b x C S O K B C E 4 6 V 6 j p 2 p N E S 8 I p 9 N C L 1 Y w m S M h 7 R r q M A B V W 6 S p p 6 i E 6 M M k B 9 K 8 4 R G q f p 7 I 8 G B U p P A M 5 N p y k V v J v 7 n d W P t X 7
  • J
E 1 G s q S D z Q 3 7 M k Q 7 R r A I Y J I S z S e G Y C K Z y Y r I C E t M t C m q Y E p w F r + 8 T F r V i m N X n N u z U v q q y M P R 3 A M Z X D g A u p w A w 1
  • A
g E J z / A K b 9 a T 9 W K 9 W x / z Z y V 7 R z C H 1 i f P 6 j r k U
  • =
< / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " s m Q n v 1 s S I v M B a K U y c Z P e T A 8 X Y v U = " > A A A B 8 n i c b V B N S w M x E M 3 W r 1 q / q h 6 9 B I t Q s Z S s C P Z Y 8 O K x g v 2 A 7 V K y a b Y N z W 6 W Z F Y
  • S
3 + G F w + K e P X X e P P f m L Z 7 N Y H A 4 / 3 Z p i Z F y R S G C D k 2 y l s b G 5 t 7 x R 3 S 3 v 7 B 4 d H 5 e O T j l G p Z r z N l F S 6 F 1 D D p Y h 5 G w R I 3 k s p 1 E g e T e Y 3 M 3 9 7 h P X R q j 4 E a Y J 9 y M 6 i k U
  • G
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  • P
z V H y q W R j w G J q k x n k s S 8 D O q Q T D J Z 6 V + a n h C 2 Y S O u G d p T C N u / G x x 8 g x f W G W I Q 6 V t x Y A X 6 u + J j E b G T K P A d k Y U x m b V m 4 v / e V 4 K Y c P P R J y k w G O 2 X B S m E
  • P
C 8 / / x U G j O Q E 4 t
  • U
w L e y t m Y 6
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A 5 t S y Y b g r r 6 8 T j r X d Z f U 3 Y e b S r O R x 1 F E Z + g c V Z G L b l E T 3 a M W a i O G F H p G r + j N A e f F e X c + l q F J 5 8 5 R X / g f P 4 A c 3 C P W w = = < / l a t e x i t >

Christopher O’Neill (SDSU) Computing delta sets of affine semigroups Feb 10, 2020 14 / 27

slide-79
SLIDE 79

A faster solution: dynamic programming

Fix n ∈ S = n1, . . . , nk. For each i ≤ k, φi : Z(n − ni) − → Z(n) a − → a + ei S = 6, 9, 20:

< l a t e x i t s h a 1 _ b a s e 6 4 = " y A
  • H
e 6 O H C j 8 O R 3 h l J Q l R T C X I N 8 = " > A A A B 9 X i c b V D L T g I x F L 2 D L 8 Q X 6 t J N I z H B D Z k h G H V H d O M S E 3 l E G E m n d K C h 5 m H Q 2 Z 8 B 9 u X G i M W / / F n X 9 j G W a h 4 E m a n J x z b + 7 p 8 S L O l L b t b y u 3 s r q 2 v p H f L G x t 7 + z u F f c P W i q M J a F N E v J Q d j y s K G e C N j X T n H Y i S X H g c d r 2 x t c z v / 1 I p W K h u N O T i L
  • B
H g r m M 4 K 1 k R 5 6 A d Y j 5 a P 7 c v X s 8 r R f L N k V O w V a J k 5 G S p C h S 9 + 9 Q Y h i Q M q N O F Y q a 5 j R 9 p N s N S M c D
  • t
9 G J F I z G e E i 7 h g
  • c
U O U m a e
  • p
O j H K A P m h N E 9
  • l
K q / N x I c K D U J P D O Z p l z Z u J / X j f W /
  • W
b M B H F m g
  • y
P + T H H O k Q z S p A A y Y p X x i C C a S m a y I j L D E R J u i C q Y E Z / H L y 6 R V r T h 2 x b m t l e p X W R 1 5 O I J j K I M D 5 1 C H G 2 h A E w h I e I Z X e L O e r B f r 3 f q Y j + a s b O c Q / s D 6 / A G y C Z F Q < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " 8 a m w z C u D a a J M U F O d I D w i V j h s U = " > A A A B 9 X i c b V D L T g I x F L 2 D L 8 Q X 6 t J N I z H B D Z k B j S 6 J b l x i I
  • 8
I I + m U D j R O p O 2
  • y
E T / s O N C 4 1 x 6 7 + 4 8 2 8 s w y w U P E m T k 3 P u z T 9 X s S Z r b 9 b e V W V t f W N / K b h a 3 t n d 2 9 4 v 5 B S 4 W x J L R J Q h 7 K j
  • c
V 5 U z Q p m a a 4 k K Q 4 8 T t v e + H r m t x + p V C w U d 3
  • S
U T f A Q 8 F 8 R r A 2 k M v w H q k f H R f r p 7 X T v v F k l 2 x U 6 B l 4 m S k B B k a / e J X b x C S O K B C E 4 6 V 6 j p 2 p N E S 8 I p 9 N C L 1 Y w m S M h 7 R r q M A B V W 6 S p p 6 i E 6 M M k B 9 K 8 4 R G q f p 7 I 8 G B U p P A M 5 N p y k V v J v 7 n d W P t X 7
  • J
E 1 G s q S D z Q 3 7 M k Q 7 R r A I Y J I S z S e G Y C K Z y Y r I C E t M t C m q Y E p w F r + 8 T F r V i m N X n N u z U v q q y M P R 3 A M Z X D g A u p w A w 1
  • A
g E J z / A K b 9 a T 9 W K 9 W x / z Z y V 7 R z C H 1 i f P 6 j r k U
  • =
< / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " s m Q n v 1 s S I v M B a K U y c Z P e T A 8 X Y v U = " > A A A B 8 n i c b V B N S w M x E M 3 W r 1 q / q h 6 9 B I t Q s Z S s C P Z Y 8 O K x g v 2 A 7 V K y a b Y N z W 6 W Z F Y
  • S
3 + G F w + K e P X X e P P f m L Z 7 N Y H A 4 / 3 Z p i Z F y R S G C D k 2 y l s b G 5 t 7 x R 3 S 3 v 7 B 4 d H 5 e O T j l G p Z r z N l F S 6 F 1 D D p Y h 5 G w R I 3 k s p 1 E g e T e Y 3 M 3 9 7 h P X R q j 4 E a Y J 9 y M 6 i k U
  • G
A U r e d n s C l f d G q m R y G 5 Q u p k A b x O 3 J x U U I 7 W
  • P
z V H y q W R j w G J q k x n k s S 8 D O q Q T D J Z 6 V + a n h C 2 Y S O u G d p T C N u / G x x 8 g x f W G W I Q 6 V t x Y A X 6 u + J j E b G T K P A d k Y U x m b V m 4 v / e V 4 K Y c P P R J y k w G O 2 X B S m E
  • P
C 8 / / x U G j O Q E 4 t
  • U
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A 5 t S y Y b g r r 6 8 T j r X d Z f U 3 Y e b S r O R x 1 F E Z + g c V Z G L b l E T 3 a M W a i O G F H p G r + j N A e f F e X c + l q F J 5 8 5 R X / g f P 4 A c 3 C P W w = = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " H B j d
  • I
I v W 9 G J V e 5 s y v z R r Y 5 H I K A = " > A A A B 9 X i c b V D L S s N A F L 2 p r 1 p f U Z d u B
  • t
Q N y U p i i 6 L b l x W s A 9 s Y 5 l M J + 3 Q y S T M T J Q S + h 9 u X C j i 1 n 9 x 5 9 8 4 T b P Q 1 g M D h 3 P u 5 Z 4 5 f s y Z
  • 7
z b R V W V t f W N 4 q b p a 3 t n d 9 e / + g p a J E E t
  • k
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  • U
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  • W
A B I 1 g b 6 a E X Y j 1 S A b q v 1 M 6 d 7 5 d d q p O B r R M 3 J y U I U e j b 3 / 1 B h F J Q i
  • 4
V i p r u v E 2 k u x 1 I x w O i 3 1 E k V j T M Z 4 S L u G C h x S 5 a V Z 6 i k 6 M c
  • A
B Z E T 2 i U q b 8 3 U h w q N Q l 9 M 5 m l X P R m 4 n 9 e N 9 H B p Z c y E S e a C j I / F C Q c 6 Q j N K k A D J i n R f G I I J p K Z r I i M s M R E m 6 J K p g R 3 8 c v L p F W r u k 7 V v T r 1 6 / y O
  • p
w B M d Q A R c u
  • A
4 3 I A m E J D w D K / w Z j 1 Z L 9 a 7 9 T E f L V j 5 z i H 8 g f X 5 A 6 R c k U c = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " t 2 v n x A 8 1 9 s / T F J R r r V a N D Z P E + H = " > A A A B 8 n i c b V B N S w M x E M 3 W r 1 q / q h 6 9 B I t Q s Z S s C P Z Y 8 O K x g v 2 A 7 V K y a b Y N z W 6 W Z F Y
  • S
3 + G F w + K e P X X e P P f m L Z 7 N Y H A 4 / 3 Z p i Z F y R S G C D k 2 y l s b G 5 t 7 x R 3 S 3 v 7 B 4 d H 5 e O T j l G p Z r z N l F S 6 F 1 D D p Y h 5 G w R I 3 k s p 1 E g e T e Y 3 M 3 9 7 h P X R q j 4 E a Y J 9 y M 6 i k U
  • G
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  • P
C 8 / / x U G j O Q E 4 t
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Christopher O’Neill (SDSU) Computing delta sets of affine semigroups Feb 10, 2020 14 / 27

slide-80
SLIDE 80

A faster solution: dynamic programming

Fix n ∈ S = n1, . . . , nk. For each i ≤ k, φi : Z(n − ni) − → Z(n) a − → a + ei S = 6, 9, 20:

< l a t e x i t s h a 1 _ b a s e 6 4 = " y A
  • H
e 6 O H C j 8 O R 3 h l J Q l R T C X I N 8 = " > A A A B 9 X i c b V D L T g I x F L 2 D L 8 Q X 6 t J N I z H B D Z k h G H V H d O M S E 3 l E G E m n d K C h 5 m H Q 2 Z 8 B 9 u X G i M W / / F n X 9 j G W a h 4 E m a n J x z b + 7 p 8 S L O l L b t b y u 3 s r q 2 v p H f L G x t 7 + z u F f c P W i q M J a F N E v J Q d j y s K G e C N j X T n H Y i S X H g c d r 2 x t c z v / 1 I p W K h u N O T i L
  • B
H g r m M 4 K 1 k R 5 6 A d Y j 5 a P 7 c v X s 8 r R f L N k V O w V a J k 5 G S p C h S 9 + 9 Q Y h i Q M q N O F Y q a 5 j R 9 p N s N S M c D
  • t
9 G J F I z G e E i 7 h g
  • c
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  • p
O j H K A P m h N E 9
  • l
K q / N x I c K D U J P D O Z p l z Z u J / X j f W /
  • W
b M B H F m g
  • y
P + T H H O k Q z S p A A y Y p X x i C C a S m a y I j L D E R J u i C q Y E Z / H L y 6 R V r T h 2 x b m t l e p X W R 1 5 O I J j K I M D 5 1 C H G 2 h A E w h I e I Z X e L O e r B f r 3 f q Y j + a s b O c Q / s D 6 / A G y C Z F Q < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " 8 a m w z C u D a a J M U F O d I D w i V j h s U = " > A A A B 9 X i c b V D L T g I x F L 2 D L 8 Q X 6 t J N I z H B D Z k B j S 6 J b l x i I
  • 8
I I + m U D j R O p O 2
  • y
E T / s O N C 4 1 x 6 7 + 4 8 2 8 s w y w U P E m T k 3 P u z T 9 X s S Z r b 9 b e V W V t f W N / K b h a 3 t n d 2 9 4 v 5 B S 4 W x J L R J Q h 7 K j
  • c
V 5 U z Q p m a a 4 k K Q 4 8 T t v e + H r m t x + p V C w U d 3
  • S
U T f A Q 8 F 8 R r A 2 k M v w H q k f H R f r p 7 X T v v F k l 2 x U 6 B l 4 m S k B B k a / e J X b x C S O K B C E 4 6 V 6 j p 2 p N E S 8 I p 9 N C L 1 Y w m S M h 7 R r q M A B V W 6 S p p 6 i E 6 M M k B 9 K 8 4 R G q f p 7 I 8 G B U p P A M 5 N p y k V v J v 7 n d W P t X 7
  • J
E 1 G s q S D z Q 3 7 M k Q 7 R r A I Y J I S z S e G Y C K Z y Y r I C E t M t C m q Y E p w F r + 8 T F r V i m N X n N u z U v q q y M P R 3 A M Z X D g A u p w A w 1
  • A
g E J z / A K b 9 a T 9 W K 9 W x / z Z y V 7 R z C H 1 i f P 6 j r k U
  • =
< / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " s m Q n v 1 s S I v M B a K U y c Z P e T A 8 X Y v U = " > A A A B 8 n i c b V B N S w M x E M 3 W r 1 q / q h 6 9 B I t Q s Z S s C P Z Y 8 O K x g v 2 A 7 V K y a b Y N z W 6 W Z F Y
  • S
3 + G F w + K e P X X e P P f m L Z 7 N Y H A 4 / 3 Z p i Z F y R S G C D k 2 y l s b G 5 t 7 x R 3 S 3 v 7 B 4 d H 5 e O T j l G p Z r z N l F S 6 F 1 D D p Y h 5 G w R I 3 k s p 1 E g e T e Y 3 M 3 9 7 h P X R q j 4 E a Y J 9 y M 6 i k U
  • G
A U r e d n s C l f d G q m R y G 5 Q u p k A b x O 3 J x U U I 7 W
  • P
z V H y q W R j w G J q k x n k s S 8 D O q Q T D J Z 6 V + a n h C 2 Y S O u G d p T C N u / G x x 8 g x f W G W I Q 6 V t x Y A X 6 u + J j E b G T K P A d k Y U x m b V m 4 v / e V 4 K Y c P P R J y k w G O 2 X B S m E
  • P
C 8 / / x U G j O Q E 4 t
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w L e y t m Y 6
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A 5 t S y Y b g r r 6 8 T j r X d Z f U 3 Y e b S r O R x 1 F E Z + g c V Z G L b l E T 3 a M W a i O G F H p G r + j N A e f F e X c + l q F J 5 8 5 R X / g f P 4 A c 3 C P W w = = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " H B j d
  • I
I v W 9 G J V e 5 s y v z R r Y 5 H I K A = " > A A A B 9 X i c b V D L S s N A F L 2 p r 1 p f U Z d u B
  • t
Q N y U p i i 6 L b l x W s A 9 s Y 5 l M J + 3 Q y S T M T J Q S + h 9 u X C j i 1 n 9 x 5 9 8 4 T b P Q 1 g M D h 3 P u 5 Z 4 5 f s y Z
  • 7
z b R V W V t f W N 4 q b p a 3 t n d 9 e / + g p a J E E t
  • k
E Y 9 k x 8 e K c i Z
  • U
z P N a S e W F I c + p 2 1 / f D 3 z 2 4 9 U K h a J O z 2 J q R f i
  • W
A B I 1 g b 6 a E X Y j 1 S A b q v 1 M 6 d 7 5 d d q p O B r R M 3 J y U I U e j b 3 / 1 B h F J Q i
  • 4
V i p r u v E 2 k u x 1 I x w O i 3 1 E k V j T M Z 4 S L u G C h x S 5 a V Z 6 i k 6 M c
  • A
B Z E T 2 i U q b 8 3 U h w q N Q l 9 M 5 m l X P R m 4 n 9 e N 9 H B p Z c y E S e a C j I / F C Q c 6 Q j N K k A D J i n R f G I I J p K Z r I i M s M R E m 6 J K p g R 3 8 c v L p F W r u k 7 V v T r 1 6 / y O
  • p
w B M d Q A R c u
  • A
4 3 I A m E J D w D K / w Z j 1 Z L 9 a 7 9 T E f L V j 5 z i H 8 g f X 5 A 6 R c k U c = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " t 2 v n x A 8 1 9 s / T F J R r r V a N D Z P E + H = " > A A A B 8 n i c b V B N S w M x E M 3 W r 1 q / q h 6 9 B I t Q s Z S s C P Z Y 8 O K x g v 2 A 7 V K y a b Y N z W 6 W Z F Y
  • S
3 + G F w + K e P X X e P P f m L Z 7 N Y H A 4 / 3 Z p i Z F y R S G C D k 2 y l s b G 5 t 7 x R 3 S 3 v 7 B 4 d H 5 e O T j l G p Z r z N l F S 6 F 1 D D p Y h 5 G w R I 3 k s p 1 E g e T e Y 3 M 3 9 7 h P X R q j 4 E a Y J 9 y M 6 i k U
  • G
A U r e d n s C l d J z a 2 R y G 5 Q u p k A b x O 3 J x U U I 7 W
  • P
z V H y q W R j w G J q k x n k s S 8 D O q Q T D J Z 6 V + a n h C 2 Y S O u G d p T C N u / G x x 8 g x f W G W I Q 6 V t x Y A X 6 u + J j E b G T K P A d k Y U x m b V m 4 v / e V 4 K Y c P P R J y k w G O 2 X B S m E
  • P
C 8 / / x U G j O Q E 4 t
  • U
w L e y t m Y 6
  • p
A 5 t S y Y b g r r 6 8 T j r X d Z f U 3 Y e b S r O R x 1 F E Z + g c V Z G L b l E T 3 a M W a i O G F H p G r + j N A e f F e X c + l q F J 5 8 5 R X / g f P 4 A c 2 6 P W w = = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " i N d x g B A 2 G B 6 W e P D 9 u c w 3 b m G 1 r 6 = " > A A A B 9 X i c b V D L T g I x F L 2 D L 8 Q X 6 t J N I z H B D Z k B E 3 V H d O M S E 3 l E G E m n d K C h 5 m H Q 2 Z 8 B 9 u X G i M W / / F n X 9 j G W a h 4 E m a n J x z b + 7 p 8 S L O l L b t b y u 3 s r q 2 v p H f L G x t 7 + z u F f c P W i q M J a F N E v J Q d j y s K G e C N j X T n H Y i S X H g c d r 2 x t c z v / 1 I p W K h u N O T i L
  • B
H g r m M 4 K 1 k R 5 6 A d Y j 5 a P 7 c r V 2 e d
  • v
l u y K n Q I t E y c j J c j Q 6 B e / e
  • O
Q x A E V m n C s V N e x I + m W G p G O J W e r G i E S Z j P K R d Q w U O q H K T N P U U n R h l g P x Q m i c S t X f G w k O l J
  • E
n p l M U y 5 6 M / E / r x t r / 8 J N m I h i T Q W Z H / J j j n S I Z h W g A Z O U a D 4 x B B P J T F Z E R l h i
  • k
1 R B V O C s / j l Z d K q V h y 7 4 t y e l e p X W R 1 5 O I J j K I M D 5 1 C H G 2 h A E w h I e I Z X e L O e r B f r 3 f q Y j + a s b O c Q / s D 6 / A G u / Z F O < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " A 2 J j 5 i 4 H O s n V Q L V f 3 m D a K v I s 9 L = " > A A A B 8 n i c b V B N S w M x E M 3 W r 1 q / q h 6 9 B I t Q s Z S s C P Z Y 8 O K x g v 2 A 7 V K y a b Y N z W 6 W Z F Y
  • S
3 + G F w + K e P X X e P P f m L Z 7 N Y H A 4 / 3 Z p i Z F y R S G C D k 2 y l s b G 5 t 7 x R 3 S 3 v 7 B 4 d H 5 e O T j l G p Z r z N l F S 6 F 1 D D p Y h 5 G w R I 3 k s p 1 E g e T e Y 3 M 3 9 7 h P X R q j 4 E a Y J 9 y M 6 i k U
  • G
A U r e d n s C l d J j d T c y G 5 Q u p k A b x O 3 J x U U I 7 W
  • P
z V H y q W R j w G J q k x n k s S 8 D O q Q T D J Z 6 V + a n h C 2 Y S O u G d p T C N u / G x x 8 g x f W G W I Q 6 V t x Y A X 6 u + J j E b G T K P A d k Y U x m b V m 4 v / e V 4 K Y c P P R J y k w G O 2 X B S m E
  • P
C 8 / / x U G j O Q E 4 t
  • U
w L e y t m Y 6
  • p
A 5 t S y Y b g r r 6 8 T j r X d Z f U 3 Y e b S r O R x 1 F E Z + g c V Z G L b l E T 3 a M W a i O G F H p G r + j N A e f F e X c + l q F J 5 8 5 R X / g f P 4 A c 2 y P W w = = < / l a t e x i t >

Christopher O’Neill (SDSU) Computing delta sets of affine semigroups Feb 10, 2020 14 / 27

slide-81
SLIDE 81

A faster solution: dynamic programming

Fix n ∈ S = n1, . . . , nk. For each i ≤ k, φi : Z(n − ni) − → Z(n) a − → a + ei Z(n) =

i≤k φi(Z(n − ni))

S = 6, 9, 20:

< l a t e x i t s h a 1 _ b a s e 6 4 = " y A
  • H
e 6 O H C j 8 O R 3 h l J Q l R T C X I N 8 = " > A A A B 9 X i c b V D L T g I x F L 2 D L 8 Q X 6 t J N I z H B D Z k h G H V H d O M S E 3 l E G E m n d K C h 5 m H Q 2 Z 8 B 9 u X G i M W / / F n X 9 j G W a h 4 E m a n J x z b + 7 p 8 S L O l L b t b y u 3 s r q 2 v p H f L G x t 7 + z u F f c P W i q M J a F N E v J Q d j y s K G e C N j X T n H Y i S X H g c d r 2 x t c z v / 1 I p W K h u N O T i L
  • B
H g r m M 4 K 1 k R 5 6 A d Y j 5 a P 7 c v X s 8 r R f L N k V O w V a J k 5 G S p C h S 9 + 9 Q Y h i Q M q N O F Y q a 5 j R 9 p N s N S M c D
  • t
9 G J F I z G e E i 7 h g
  • c
U O U m a e
  • p
O j H K A P m h N E 9
  • l
K q / N x I c K D U J P D O Z p l z Z u J / X j f W /
  • W
b M B H F m g
  • y
P + T H H O k Q z S p A A y Y p X x i C C a S m a y I j L D E R J u i C q Y E Z / H L y 6 R V r T h 2 x b m t l e p X W R 1 5 O I J j K I M D 5 1 C H G 2 h A E w h I e I Z X e L O e r B f r 3 f q Y j + a s b O c Q / s D 6 / A G y C Z F Q < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " 8 a m w z C u D a a J M U F O d I D w i V j h s U = " > A A A B 9 X i c b V D L T g I x F L 2 D L 8 Q X 6 t J N I z H B D Z k B j S 6 J b l x i I
  • 8
I I + m U D j R O p O 2
  • y
E T / s O N C 4 1 x 6 7 + 4 8 2 8 s w y w U P E m T k 3 P u z T 9 X s S Z r b 9 b e V W V t f W N / K b h a 3 t n d 2 9 4 v 5 B S 4 W x J L R J Q h 7 K j
  • c
V 5 U z Q p m a a 4 k K Q 4 8 T t v e + H r m t x + p V C w U d 3
  • S
U T f A Q 8 F 8 R r A 2 k M v w H q k f H R f r p 7 X T v v F k l 2 x U 6 B l 4 m S k B B k a / e J X b x C S O K B C E 4 6 V 6 j p 2 p N E S 8 I p 9 N C L 1 Y w m S M h 7 R r q M A B V W 6 S p p 6 i E 6 M M k B 9 K 8 4 R G q f p 7 I 8 G B U p P A M 5 N p y k V v J v 7 n d W P t X 7
  • J
E 1 G s q S D z Q 3 7 M k Q 7 R r A I Y J I S z S e G Y C K Z y Y r I C E t M t C m q Y E p w F r + 8 T F r V i m N X n N u z U v q q y M P R 3 A M Z X D g A u p w A w 1
  • A
g E J z / A K b 9 a T 9 W K 9 W x / z Z y V 7 R z C H 1 i f P 6 j r k U
  • =
< / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " s m Q n v 1 s S I v M B a K U y c Z P e T A 8 X Y v U = " > A A A B 8 n i c b V B N S w M x E M 3 W r 1 q / q h 6 9 B I t Q s Z S s C P Z Y 8 O K x g v 2 A 7 V K y a b Y N z W 6 W Z F Y
  • S
3 + G F w + K e P X X e P P f m L Z 7 N Y H A 4 / 3 Z p i Z F y R S G C D k 2 y l s b G 5 t 7 x R 3 S 3 v 7 B 4 d H 5 e O T j l G p Z r z N l F S 6 F 1 D D p Y h 5 G w R I 3 k s p 1 E g e T e Y 3 M 3 9 7 h P X R q j 4 E a Y J 9 y M 6 i k U
  • G
A U r e d n s C l f d G q m R y G 5 Q u p k A b x O 3 J x U U I 7 W
  • P
z V H y q W R j w G J q k x n k s S 8 D O q Q T D J Z 6 V + a n h C 2 Y S O u G d p T C N u / G x x 8 g x f W G W I Q 6 V t x Y A X 6 u + J j E b G T K P A d k Y U x m b V m 4 v / e V 4 K Y c P P R J y k w G O 2 X B S m E
  • P
C 8 / / x U G j O Q E 4 t
  • U
w L e y t m Y 6
  • p
A 5 t S y Y b g r r 6 8 T j r X d Z f U 3 Y e b S r O R x 1 F E Z + g c V Z G L b l E T 3 a M W a i O G F H p G r + j N A e f F e X c + l q F J 5 8 5 R X / g f P 4 A c 3 C P W w = = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " H B j d
  • I
I v W 9 G J V e 5 s y v z R r Y 5 H I K A = " > A A A B 9 X i c b V D L S s N A F L 2 p r 1 p f U Z d u B
  • t
Q N y U p i i 6 L b l x W s A 9 s Y 5 l M J + 3 Q y S T M T J Q S + h 9 u X C j i 1 n 9 x 5 9 8 4 T b P Q 1 g M D h 3 P u 5 Z 4 5 f s y Z
  • 7
z b R V W V t f W N 4 q b p a 3 t n d 9 e / + g p a J E E t
  • k
E Y 9 k x 8 e K c i Z
  • U
z P N a S e W F I c + p 2 1 / f D 3 z 2 4 9 U K h a J O z 2 J q R f i
  • W
A B I 1 g b 6 a E X Y j 1 S A b q v 1 M 6 d 7 5 d d q p O B r R M 3 J y U I U e j b 3 / 1 B h F J Q i
  • 4
V i p r u v E 2 k u x 1 I x w O i 3 1 E k V j T M Z 4 S L u G C h x S 5 a V Z 6 i k 6 M c
  • A
B Z E T 2 i U q b 8 3 U h w q N Q l 9 M 5 m l X P R m 4 n 9 e N 9 H B p Z c y E S e a C j I / F C Q c 6 Q j N K k A D J i n R f G I I J p K Z r I i M s M R E m 6 J K p g R 3 8 c v L p F W r u k 7 V v T r 1 6 / y O
  • p
w B M d Q A R c u
  • A
4 3 I A m E J D w D K / w Z j 1 Z L 9 a 7 9 T E f L V j 5 z i H 8 g f X 5 A 6 R c k U c = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " t 2 v n x A 8 1 9 s / T F J R r r V a N D Z P E + H = " > A A A B 8 n i c b V B N S w M x E M 3 W r 1 q / q h 6 9 B I t Q s Z S s C P Z Y 8 O K x g v 2 A 7 V K y a b Y N z W 6 W Z F Y
  • S
3 + G F w + K e P X X e P P f m L Z 7 N Y H A 4 / 3 Z p i Z F y R S G C D k 2 y l s b G 5 t 7 x R 3 S 3 v 7 B 4 d H 5 e O T j l G p Z r z N l F S 6 F 1 D D p Y h 5 G w R I 3 k s p 1 E g e T e Y 3 M 3 9 7 h P X R q j 4 E a Y J 9 y M 6 i k U
  • G
A U r e d n s C l d J z a 2 R y G 5 Q u p k A b x O 3 J x U U I 7 W
  • P
z V H y q W R j w G J q k x n k s S 8 D O q Q T D J Z 6 V + a n h C 2 Y S O u G d p T C N u / G x x 8 g x f W G W I Q 6 V t x Y A X 6 u + J j E b G T K P A d k Y U x m b V m 4 v / e V 4 K Y c P P R J y k w G O 2 X B S m E
  • P
C 8 / / x U G j O Q E 4 t
  • U
w L e y t m Y 6
  • p
A 5 t S y Y b g r r 6 8 T j r X d Z f U 3 Y e b S r O R x 1 F E Z + g c V Z G L b l E T 3 a M W a i O G F H p G r + j N A e f F e X c + l q F J 5 8 5 R X / g f P 4 A c 2 6 P W w = = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " i N d x g B A 2 G B 6 W e P D 9 u c w 3 b m G 1 r 6 = " > A A A B 9 X i c b V D L T g I x F L 2 D L 8 Q X 6 t J N I z H B D Z k B E 3 V H d O M S E 3 l E G E m n d K C h 5 m H Q 2 Z 8 B 9 u X G i M W / / F n X 9 j G W a h 4 E m a n J x z b + 7 p 8 S L O l L b t b y u 3 s r q 2 v p H f L G x t 7 + z u F f c P W i q M J a F N E v J Q d j y s K G e C N j X T n H Y i S X H g c d r 2 x t c z v / 1 I p W K h u N O T i L
  • B
H g r m M 4 K 1 k R 5 6 A d Y j 5 a P 7 c r V 2 e d
  • v
l u y K n Q I t E y c j J c j Q 6 B e / e
  • O
Q x A E V m n C s V N e x I + m W G p G O J W e r G i E S Z j P K R d Q w U O q H K T N P U U n R h l g P x Q m i c S t X f G w k O l J
  • E
n p l M U y 5 6 M / E / r x t r / 8 J N m I h i T Q W Z H / J j j n S I Z h W g A Z O U a D 4 x B B P J T F Z E R l h i
  • k
1 R B V O C s / j l Z d K q V h y 7 4 t y e l e p X W R 1 5 O I J j K I M D 5 1 C H G 2 h A E w h I e I Z X e L O e r B f r 3 f q Y j + a s b O c Q / s D 6 / A G u / Z F O < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " A 2 J j 5 i 4 H O s n V Q L V f 3 m D a K v I s 9 L = " > A A A B 8 n i c b V B N S w M x E M 3 W r 1 q / q h 6 9 B I t Q s Z S s C P Z Y 8 O K x g v 2 A 7 V K y a b Y N z W 6 W Z F Y
  • S
3 + G F w + K e P X X e P P f m L Z 7 N Y H A 4 / 3 Z p i Z F y R S G C D k 2 y l s b G 5 t 7 x R 3 S 3 v 7 B 4 d H 5 e O T j l G p Z r z N l F S 6 F 1 D D p Y h 5 G w R I 3 k s p 1 E g e T e Y 3 M 3 9 7 h P X R q j 4 E a Y J 9 y M 6 i k U
  • G
A U r e d n s C l d J j d T c y G 5 Q u p k A b x O 3 J x U U I 7 W
  • P
z V H y q W R j w G J q k x n k s S 8 D O q Q T D J Z 6 V + a n h C 2 Y S O u G d p T C N u / G x x 8 g x f W G W I Q 6 V t x Y A X 6 u + J j E b G T K P A d k Y U x m b V m 4 v / e V 4 K Y c P P R J y k w G O 2 X B S m E
  • P
C 8 / / x U G j O Q E 4 t
  • U
w L e y t m Y 6
  • p
A 5 t S y Y b g r r 6 8 T j r X d Z f U 3 Y e b S r O R x 1 F E Z + g c V Z G L b l E T 3 a M W a i O G F H p G r + j N A e f F e X c + l q F J 5 8 5 R X / g f P 4 A c 2 y P W w = = < / l a t e x i t >

Christopher O’Neill (SDSU) Computing delta sets of affine semigroups Feb 10, 2020 14 / 27

slide-82
SLIDE 82

A faster solution: dynamic programming

Fix n ∈ S = n1, . . . , nk. For each i ≤ k, φi : Z(n − ni) − → Z(n) a − → a + ei Z(n) =

i≤k φi(Z(n − ni))

Christopher O’Neill (SDSU) Computing delta sets of affine semigroups Feb 10, 2020 14 / 27

slide-83
SLIDE 83

A faster solution: dynamic programming

Fix n ∈ S = n1, . . . , nk. For each i ≤ k, φi : Z(n − ni) − → Z(n) a − → a + ei Z(n) =

i≤k φi(Z(n − ni))

n ∈ S = 6, 9, 20 Z(n) L(n) {0} {0}

Christopher O’Neill (SDSU) Computing delta sets of affine semigroups Feb 10, 2020 14 / 27

slide-84
SLIDE 84

A faster solution: dynamic programming

Fix n ∈ S = n1, . . . , nk. For each i ≤ k, φi : Z(n − ni) − → Z(n) a − → a + ei Z(n) =

i≤k φi(Z(n − ni))

n ∈ S = 6, 9, 20 Z(n) L(n) {0} {0} 6

6

e1 {e1} {1}

Christopher O’Neill (SDSU) Computing delta sets of affine semigroups Feb 10, 2020 14 / 27

slide-85
SLIDE 85

A faster solution: dynamic programming

Fix n ∈ S = n1, . . . , nk. For each i ≤ k, φi : Z(n − ni) − → Z(n) a − → a + ei Z(n) =

i≤k φi(Z(n − ni))

n ∈ S = 6, 9, 20 Z(n) L(n) {0} {0} 6

6

e1 {e1} {1} 9

9

e2 {e2} {1}

Christopher O’Neill (SDSU) Computing delta sets of affine semigroups Feb 10, 2020 14 / 27

slide-86
SLIDE 86

A faster solution: dynamic programming

Fix n ∈ S = n1, . . . , nk. For each i ≤ k, φi : Z(n − ni) − → Z(n) a − → a + ei Z(n) =

i≤k φi(Z(n − ni))

n ∈ S = 6, 9, 20 Z(n) L(n) {0} {0} 6

6

e1 {e1} {1} 9

9

e2 {e2} {1} 12 e1

6

2e1 {2e1} {2}

Christopher O’Neill (SDSU) Computing delta sets of affine semigroups Feb 10, 2020 14 / 27

slide-87
SLIDE 87

A faster solution: dynamic programming

Fix n ∈ S = n1, . . . , nk. For each i ≤ k, φi : Z(n − ni) − → Z(n) a − → a + ei Z(n) =

i≤k φi(Z(n − ni))

n ∈ S = 6, 9, 20 Z(n) L(n) {0} {0} 6

6

e1 {e1} {1} 9

9

e2 {e2} {1} 12 e1

6

2e1 {2e1} {2} 15 e2

6

(1, 1, 0) {(1, 1, 0)} {2}

Christopher O’Neill (SDSU) Computing delta sets of affine semigroups Feb 10, 2020 14 / 27

slide-88
SLIDE 88

A faster solution: dynamic programming

Fix n ∈ S = n1, . . . , nk. For each i ≤ k, φi : Z(n − ni) − → Z(n) a − → a + ei Z(n) =

i≤k φi(Z(n − ni))

n ∈ S = 6, 9, 20 Z(n) L(n) {0} {0} 6

6

e1 {e1} {1} 9

9

e2 {e2} {1} 12 e1

6

2e1 {2e1} {2} 15 e2

6

(1, 1, 0) {(1, 1, 0)} {2} e1

9

(1, 1, 0)

Christopher O’Neill (SDSU) Computing delta sets of affine semigroups Feb 10, 2020 14 / 27

slide-89
SLIDE 89

A faster solution: dynamic programming

Fix n ∈ S = n1, . . . , nk. For each i ≤ k, φi : Z(n − ni) − → Z(n) a − → a + ei Z(n) =

i≤k φi(Z(n − ni))

n ∈ S = 6, 9, 20 Z(n) L(n) {0} {0} 6

6

e1 {e1} {1} 9

9

e2 {e2} {1} 12 e1

6

2e1 {2e1} {2} 15 e2

6

(1, 1, 0) {(1, 1, 0)} {2} e1

9

(1, 1, 0) 18 2e1

6

3e1 {3e1, 2e2} {2, 3}

Christopher O’Neill (SDSU) Computing delta sets of affine semigroups Feb 10, 2020 14 / 27

slide-90
SLIDE 90

A faster solution: dynamic programming

Fix n ∈ S = n1, . . . , nk. For each i ≤ k, φi : Z(n − ni) − → Z(n) a − → a + ei Z(n) =

i≤k φi(Z(n − ni))

n ∈ S = 6, 9, 20 Z(n) L(n) {0} {0} 6

6

e1 {e1} {1} 9

9

e2 {e2} {1} 12 e1

6

2e1 {2e1} {2} 15 e2

6

(1, 1, 0) {(1, 1, 0)} {2} e1

9

(1, 1, 0) 18 2e1

6

3e1 {3e1, 2e2} {2, 3} e2

9

2e2

Christopher O’Neill (SDSU) Computing delta sets of affine semigroups Feb 10, 2020 14 / 27

slide-91
SLIDE 91

A faster solution: dynamic programming

Fix n ∈ S = n1, . . . , nk. For each i ≤ k, φi : Z(n − ni) − → Z(n) a − → a + ei Z(n) =

i≤k φi(Z(n − ni))

n ∈ S = 6, 9, 20 Z(n) L(n) {0} {0} 6

6

e1 {e1} {1} 9

9

e2 {e2} {1} 12 e1

6

2e1 {2e1} {2} 15 e2

6

(1, 1, 0) {(1, 1, 0)} {2} e1

9

(1, 1, 0) 18 2e1

6

3e1 {3e1, 2e2} {2, 3} e2

9

2e2 20

20

e3 {e3} {1}

Christopher O’Neill (SDSU) Computing delta sets of affine semigroups Feb 10, 2020 14 / 27

slide-92
SLIDE 92

A faster solution: dynamic programming

Fix n ∈ S = n1, . . . , nk. For each i ≤ k, φi : Z(n − ni) − → Z(n) a − → a + ei Z(n) =

i≤k φi(Z(n − ni))

n ∈ S = 6, 9, 20 Z(n) L(n) {0} {0} 6

6

e1 {e1} {1} 9

9

e2 {e2} {1} 12 e1

6

2e1 {2e1} {2} 15 e2

6

(1, 1, 0) {(1, 1, 0)} {2} e1

9

(1, 1, 0) 18 2e1

6

3e1 {3e1, 2e2} {2, 3} e2

9

2e2 20

20

e3 {e3} {1} . . . . . . . . .

Christopher O’Neill (SDSU) Computing delta sets of affine semigroups Feb 10, 2020 14 / 27

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SLIDE 93

A faster solution: dynamic programming

Fix n ∈ S = n1, . . . , nk. For each i ≤ k, φi : Z(n − ni) − → Z(n) ψi : L(n − ni) − → L(n) a − → a + ei Z(n) =

i≤k φi(Z(n − ni))

n ∈ S = 6, 9, 20 Z(n) L(n) {0} {0} 6

6

e1 {e1} {1} 9

9

e2 {e2} {1} 12 e1

6

2e1 {2e1} {2} 15 e2

6

(1, 1, 0) {(1, 1, 0)} {2} e1

9

(1, 1, 0) 18 2e1

6

3e1 {3e1, 2e2} {2, 3} e2

9

2e2 20

20

e3 {e3} {1} . . . . . . . . .

Christopher O’Neill (SDSU) Computing delta sets of affine semigroups Feb 10, 2020 14 / 27

slide-94
SLIDE 94

A faster solution: dynamic programming

Fix n ∈ S = n1, . . . , nk. For each i ≤ k, φi : Z(n − ni) − → Z(n) ψi : L(n − ni) − → L(n) a − → a + ei ℓ − → ℓ + 1 Z(n) =

i≤k φi(Z(n − ni))

n ∈ S = 6, 9, 20 Z(n) L(n) {0} {0} 6

6

e1 {e1} {1} 9

9

e2 {e2} {1} 12 e1

6

2e1 {2e1} {2} 15 e2

6

(1, 1, 0) {(1, 1, 0)} {2} e1

9

(1, 1, 0) 18 2e1

6

3e1 {3e1, 2e2} {2, 3} e2

9

2e2 20

20

e3 {e3} {1} . . . . . . . . .

Christopher O’Neill (SDSU) Computing delta sets of affine semigroups Feb 10, 2020 14 / 27

slide-95
SLIDE 95

A faster solution: dynamic programming

Fix n ∈ S = n1, . . . , nk. For each i ≤ k, φi : Z(n − ni) − → Z(n) ψi : L(n − ni) − → L(n) a − → a + ei ℓ − → ℓ + 1 Z(n) =

i≤k φi(Z(n − ni))

L(n) =

i≤k ψi(L(n − ni))

n ∈ S = 6, 9, 20 Z(n) L(n) {0} {0} 6

6

e1 {e1} {1} 9

9

e2 {e2} {1} 12 e1

6

2e1 {2e1} {2} 15 e2

6

(1, 1, 0) {(1, 1, 0)} {2} e1

9

(1, 1, 0) 18 2e1

6

3e1 {3e1, 2e2} {2, 3} e2

9

2e2 20

20

e3 {e3} {1} . . . . . . . . .

Christopher O’Neill (SDSU) Computing delta sets of affine semigroups Feb 10, 2020 14 / 27

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SLIDE 96

A faster solution: dynamic programming

Fix n ∈ S = n1, . . . , nk. For each i ≤ k, φi : Z(n − ni) − → Z(n) ψi : L(n − ni) − → L(n) a − → a + ei ℓ − → ℓ + 1 Z(n) =

i≤k φi(Z(n − ni))

L(n) =

i≤k ψi(L(n − ni))

n ∈ S = 6, 9, 20 L(n) {0} 6 9 12 15 18 20 . . .

Christopher O’Neill (SDSU) Computing delta sets of affine semigroups Feb 10, 2020 14 / 27

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SLIDE 97

A faster solution: dynamic programming

Fix n ∈ S = n1, . . . , nk. For each i ≤ k, φi : Z(n − ni) − → Z(n) ψi : L(n − ni) − → L(n) a − → a + ei ℓ − → ℓ + 1 Z(n) =

i≤k φi(Z(n − ni))

L(n) =

i≤k ψi(L(n − ni))

n ∈ S = 6, 9, 20 L(n) {0} 6 {1}

6

1 9 12 15 18 20 . . .

Christopher O’Neill (SDSU) Computing delta sets of affine semigroups Feb 10, 2020 14 / 27

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SLIDE 98

A faster solution: dynamic programming

Fix n ∈ S = n1, . . . , nk. For each i ≤ k, φi : Z(n − ni) − → Z(n) ψi : L(n − ni) − → L(n) a − → a + ei ℓ − → ℓ + 1 Z(n) =

i≤k φi(Z(n − ni))

L(n) =

i≤k ψi(L(n − ni))

n ∈ S = 6, 9, 20 L(n) {0} 6 {1}

6

1 9 {1}

9

1 12 15 18 20 . . .

Christopher O’Neill (SDSU) Computing delta sets of affine semigroups Feb 10, 2020 14 / 27

slide-99
SLIDE 99

A faster solution: dynamic programming

Fix n ∈ S = n1, . . . , nk. For each i ≤ k, φi : Z(n − ni) − → Z(n) ψi : L(n − ni) − → L(n) a − → a + ei ℓ − → ℓ + 1 Z(n) =

i≤k φi(Z(n − ni))

L(n) =

i≤k ψi(L(n − ni))

n ∈ S = 6, 9, 20 L(n) {0} 6 {1}

6

1 9 {1}

9

1 12 {2} 1

6

2 15 18 20 . . .

Christopher O’Neill (SDSU) Computing delta sets of affine semigroups Feb 10, 2020 14 / 27

slide-100
SLIDE 100

A faster solution: dynamic programming

Fix n ∈ S = n1, . . . , nk. For each i ≤ k, φi : Z(n − ni) − → Z(n) ψi : L(n − ni) − → L(n) a − → a + ei ℓ − → ℓ + 1 Z(n) =

i≤k φi(Z(n − ni))

L(n) =

i≤k ψi(L(n − ni))

n ∈ S = 6, 9, 20 L(n) {0} 6 {1}

6

1 9 {1}

9

1 12 {2} 1

6

2 15 {2} 1

6

2 18 20 . . .

Christopher O’Neill (SDSU) Computing delta sets of affine semigroups Feb 10, 2020 14 / 27

slide-101
SLIDE 101

A faster solution: dynamic programming

Fix n ∈ S = n1, . . . , nk. For each i ≤ k, φi : Z(n − ni) − → Z(n) ψi : L(n − ni) − → L(n) a − → a + ei ℓ − → ℓ + 1 Z(n) =

i≤k φi(Z(n − ni))

L(n) =

i≤k ψi(L(n − ni))

n ∈ S = 6, 9, 20 L(n) {0} 6 {1}

6

1 9 {1}

9

1 12 {2} 1

6

2 15 {2} 1

6

2 1

9

2 18 20 . . .

Christopher O’Neill (SDSU) Computing delta sets of affine semigroups Feb 10, 2020 14 / 27

slide-102
SLIDE 102

A faster solution: dynamic programming

Fix n ∈ S = n1, . . . , nk. For each i ≤ k, φi : Z(n − ni) − → Z(n) ψi : L(n − ni) − → L(n) a − → a + ei ℓ − → ℓ + 1 Z(n) =

i≤k φi(Z(n − ni))

L(n) =

i≤k ψi(L(n − ni))

n ∈ S = 6, 9, 20 L(n) {0} 6 {1}

6

1 9 {1}

9

1 12 {2} 1

6

2 15 {2} 1

6

2 1

9

2 18 {2, 3} 2

6

3 20 . . .

Christopher O’Neill (SDSU) Computing delta sets of affine semigroups Feb 10, 2020 14 / 27

slide-103
SLIDE 103

A faster solution: dynamic programming

Fix n ∈ S = n1, . . . , nk. For each i ≤ k, φi : Z(n − ni) − → Z(n) ψi : L(n − ni) − → L(n) a − → a + ei ℓ − → ℓ + 1 Z(n) =

i≤k φi(Z(n − ni))

L(n) =

i≤k ψi(L(n − ni))

n ∈ S = 6, 9, 20 L(n) {0} 6 {1}

6

1 9 {1}

9

1 12 {2} 1

6

2 15 {2} 1

6

2 1

9

2 18 {2, 3} 2

6

3 1

9

2 20 . . .

Christopher O’Neill (SDSU) Computing delta sets of affine semigroups Feb 10, 2020 14 / 27

slide-104
SLIDE 104

A faster solution: dynamic programming

Fix n ∈ S = n1, . . . , nk. For each i ≤ k, φi : Z(n − ni) − → Z(n) ψi : L(n − ni) − → L(n) a − → a + ei ℓ − → ℓ + 1 Z(n) =

i≤k φi(Z(n − ni))

L(n) =

i≤k ψi(L(n − ni))

n ∈ S = 6, 9, 20 L(n) {0} 6 {1}

6

1 9 {1}

9

1 12 {2} 1

6

2 15 {2} 1

6

2 1

9

2 18 {2, 3} 2

6

3 1

9

2 20 {1}

20

1 . . .

Christopher O’Neill (SDSU) Computing delta sets of affine semigroups Feb 10, 2020 14 / 27

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SLIDE 105

A faster solution: dynamic programming

Fix n ∈ S = n1, . . . , nk. For each i ≤ k, φi : Z(n − ni) − → Z(n) ψi : L(n − ni) − → L(n) a − → a + ei ℓ − → ℓ + 1 Z(n) =

i≤k φi(Z(n − ni))

L(n) =

i≤k ψi(L(n − ni))

n ∈ S = 6, 9, 20 L(n) {0} 6 {1}

6

1 9 {1}

9

1 12 {2} 1

6

2 15 {2} 1

6

2 1

9

2 18 {2, 3} 2

6

3 1

9

2 20 {1}

20

1 . . . . . . . . .

Christopher O’Neill (SDSU) Computing delta sets of affine semigroups Feb 10, 2020 14 / 27

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SLIDE 106

Computing the delta set dynamically

Theorem (Garc´ ıa-Garc´ ıa–Moreno-Fr´ ıas–Vigneron-Tenorio, 2014)

S = n1, . . . , nk. For n ≥ NS, ∆(n) = ∆(n + lcm(n1, nk)).

Christopher O’Neill (SDSU) Computing delta sets of affine semigroups Feb 10, 2020 15 / 27

slide-107
SLIDE 107

Computing the delta set dynamically

Theorem (Garc´ ıa-Garc´ ıa–Moreno-Fr´ ıas–Vigneron-Tenorio, 2014)

S = n1, . . . , nk. For n ≥ NS, ∆(n) = ∆(n + lcm(n1, nk)). For n ∈ S with 0 ≤ n ≤ NS + lcm(n1, nk), compute: Z(n) = {a ∈ Zk

≥0 : n = a1n1 + · · · + aknk}

Z(n) L(n) L(n) ∆(n) Compute ∆(S) =

n ∆(n).

Christopher O’Neill (SDSU) Computing delta sets of affine semigroups Feb 10, 2020 15 / 27

slide-108
SLIDE 108

Computing the delta set dynamically

Theorem (Garc´ ıa-Garc´ ıa–Moreno-Fr´ ıas–Vigneron-Tenorio, 2014)

S = n1, . . . , nk. For n ≥ NS, ∆(n) = ∆(n + lcm(n1, nk)). For n ∈ S with 0 ≤ n ≤ NS + lcm(n1, nk), compute: Z(n) = {a ∈ Zk

≥0 : n = a1n1 + · · · + aknk}

L(n − ∗) L(n) L(n) ∆(n) Compute ∆(S) =

n ∆(n).

Christopher O’Neill (SDSU) Computing delta sets of affine semigroups Feb 10, 2020 15 / 27

slide-109
SLIDE 109

Computing the delta set dynamically

Theorem (Garc´ ıa-Garc´ ıa–Moreno-Fr´ ıas–Vigneron-Tenorio, 2014)

S = n1, . . . , nk. For n ≥ NS, ∆(n) = ∆(n + lcm(n1, nk)). For n ∈ S with 0 ≤ n ≤ NS + lcm(n1, nk), compute: Z(n) = {a ∈ Zk

≥0 : n = a1n1 + · · · + aknk}

L(n − ∗) L(n) L(n) ∆(n) Compute ∆(S) =

n ∆(n).

This is significantly faster!

Christopher O’Neill (SDSU) Computing delta sets of affine semigroups Feb 10, 2020 15 / 27

slide-110
SLIDE 110

Computing the delta set dynamically

Theorem (Garc´ ıa-Garc´ ıa–Moreno-Fr´ ıas–Vigneron-Tenorio, 2014)

S = n1, . . . , nk. For n ≥ NS, ∆(n) = ∆(n + lcm(n1, nk)). For n ∈ S with 0 ≤ n ≤ NS + lcm(n1, nk), compute: Z(n) = {a ∈ Zk

≥0 : n = a1n1 + · · · + aknk}

L(n − ∗) L(n) L(n) ∆(n) Compute ∆(S) =

n ∆(n).

This is significantly faster! |Z(n)| ≈ nk−1

Christopher O’Neill (SDSU) Computing delta sets of affine semigroups Feb 10, 2020 15 / 27

slide-111
SLIDE 111

Computing the delta set dynamically

Theorem (Garc´ ıa-Garc´ ıa–Moreno-Fr´ ıas–Vigneron-Tenorio, 2014)

S = n1, . . . , nk. For n ≥ NS, ∆(n) = ∆(n + lcm(n1, nk)). For n ∈ S with 0 ≤ n ≤ NS + lcm(n1, nk), compute: Z(n) = {a ∈ Zk

≥0 : n = a1n1 + · · · + aknk}

L(n − ∗) L(n) L(n) ∆(n) Compute ∆(S) =

n ∆(n).

This is significantly faster! |Z(n)| ≈ nk−1 |L(n)| ≈ n

Christopher O’Neill (SDSU) Computing delta sets of affine semigroups Feb 10, 2020 15 / 27

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SLIDE 112

Runtime comparison

Christopher O’Neill (SDSU) Computing delta sets of affine semigroups Feb 10, 2020 16 / 27

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SLIDE 113

Runtime comparison

S NS ∆(S) Manual Dynamic 7, 15, 17, 18, 20 1935 {1, 2, 3} 1m 28s 146ms 11, 53, 73, 87 14381 {2, 4, 6, 8, 10, 22} 0m 49s 2.5s 31, 73, 77, 87, 91 31364 {2, 4, 6} 400m 12s 4.2s 100, 121, 142, 163, 284 24850 {21} ——— 0m 3.6s 1001, 1211, 1421, 1631, 2841 2063141 {10, 20, 30} ——— 1m 56s

Christopher O’Neill (SDSU) Computing delta sets of affine semigroups Feb 10, 2020 16 / 27

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SLIDE 114

Runtime comparison

S NS ∆(S) Manual Dynamic 7, 15, 17, 18, 20 1935 {1, 2, 3} 1m 28s 146ms 11, 53, 73, 87 14381 {2, 4, 6, 8, 10, 22} 0m 49s 2.5s 31, 73, 77, 87, 91 31364 {2, 4, 6} 400m 12s 4.2s 100, 121, 142, 163, 284 24850 {21} ——— 0m 3.6s 1001, 1211, 1421, 1631, 2841 2063141 {10, 20, 30} ——— 1m 56s

GAP Numerical Semigroups Package, available at http://www.gap-system.org/Packages/numericalsgps.html.

Christopher O’Neill (SDSU) Computing delta sets of affine semigroups Feb 10, 2020 16 / 27

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SLIDE 115

Generalize to affine semigroups?

Key obstruction: what does “eventually periodic” mean?

Christopher O’Neill (SDSU) Computing delta sets of affine semigroups Feb 10, 2020 17 / 27

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SLIDE 116

Generalize to affine semigroups?

Key obstruction: what does “eventually periodic” mean? Example: S = (1, 1), (1, 5), (2, 5), (3, 5), (5, 1), (5, 2), (5, 3) ⊂ Z2

≥0

Christopher O’Neill (SDSU) Computing delta sets of affine semigroups Feb 10, 2020 17 / 27

slide-117
SLIDE 117

Generalize to affine semigroups?

Key obstruction: what does “eventually periodic” mean? Example: S = (1, 1), (1, 5), (2, 5), (3, 5), (5, 1), (5, 2), (5, 3) ⊂ Z2

≥0

1 ∈ ∆(S) 2 ∈ ∆(S) ∆(S) = {1, 2, 4}

Christopher O’Neill (SDSU) Computing delta sets of affine semigroups Feb 10, 2020 17 / 27

slide-118
SLIDE 118

Generalize to affine semigroups?

Key obstruction: what does “eventually periodic” mean? Example: S = (1, 1), (1, 5), (2, 5), (3, 5), (5, 1), (5, 2), (5, 3) ⊂ Z2

≥0

1 ∈ ∆(S) 2 ∈ ∆(S) ∆(S) = {1, 2, 4} Need a new approach!

Christopher O’Neill (SDSU) Computing delta sets of affine semigroups Feb 10, 2020 17 / 27

slide-119
SLIDE 119

Commutative algebra hiding in the background

Fix an affine semigroup S = n1, . . . , nk ⊂ Zd

≥0.

n = a1n1 + · · · + aknk

  • a = (a1, . . . , ak) ∈ Zk

≥0

Christopher O’Neill (SDSU) Computing delta sets of affine semigroups Feb 10, 2020 18 / 27

slide-120
SLIDE 120

Commutative algebra hiding in the background

Fix an affine semigroup S = n1, . . . , nk ⊂ Zd

≥0.

n = a1n1 + · · · + aknk

  • a = (a1, . . . , ak) ∈ Zk

≥0

Factorization homomorphism: π : Zk

≥0

− → n1, . . . , nk a − → a1n1 + · · · + aknk

Christopher O’Neill (SDSU) Computing delta sets of affine semigroups Feb 10, 2020 18 / 27

slide-121
SLIDE 121

Commutative algebra hiding in the background

Fix an affine semigroup S = n1, . . . , nk ⊂ Zd

≥0.

n = a1n1 + · · · + aknk

  • a = (a1, . . . , ak) ∈ Zk

≥0

Factorization homomorphism: π : Zk

≥0

− → n1, . . . , nk a − → a1n1 + · · · + aknk

Definition

The kernel ker π is the relation ∼ on Zk

≥0 with a ∼ b whenever

π(a) = π(b)

Christopher O’Neill (SDSU) Computing delta sets of affine semigroups Feb 10, 2020 18 / 27

slide-122
SLIDE 122

Commutative algebra hiding in the background

Fix an affine semigroup S = n1, . . . , nk ⊂ Zd

≥0.

n = a1n1 + · · · + aknk

  • a = (a1, . . . , ak) ∈ Zk

≥0

Factorization homomorphism: π : Zk

≥0

− → n1, . . . , nk a − → a1n1 + · · · + aknk

Definition

The kernel ker π is the relation ∼ on Zk

≥0 with a ∼ b whenever

π(a) = π(b) ker π is a congruence: an equivalence relation a ∼ a a ∼ b ⇒ b ∼ a a ∼ b and b ∼ c ⇒ a ∼ c

Christopher O’Neill (SDSU) Computing delta sets of affine semigroups Feb 10, 2020 18 / 27

slide-123
SLIDE 123

Commutative algebra hiding in the background

Fix an affine semigroup S = n1, . . . , nk ⊂ Zd

≥0.

n = a1n1 + · · · + aknk

  • a = (a1, . . . , ak) ∈ Zk

≥0

Factorization homomorphism: π : Zk

≥0

− → n1, . . . , nk a − → a1n1 + · · · + aknk

Definition

The kernel ker π is the relation ∼ on Zk

≥0 with a ∼ b whenever

π(a) = π(b) ker π is a congruence: an equivalence relation a ∼ a a ∼ b ⇒ b ∼ a a ∼ b and b ∼ c ⇒ a ∼ c that is closed under translation. a ∼ b ⇒ a + c ∼ b + c

Christopher O’Neill (SDSU) Computing delta sets of affine semigroups Feb 10, 2020 18 / 27

slide-124
SLIDE 124

Commutative algebra hiding in the background

Fix an affine semigroup S = n1, . . . , nk ⊂ Zd

≥0.

n = a1n1 + · · · + aknk

  • a = (a1, . . . , ak) ∈ Zk

≥0

Factorization homomorphism: π : Zk

≥0

− → n1, . . . , nk a − → a1n1 + · · · + aknk

Definition

The kernel ker π is the relation ∼ on Zk

≥0 with a ∼ b whenever

π(a) = π(b) ker π is a congruence: an equivalence relation a ∼ a a ∼ b ⇒ b ∼ a a ∼ b and b ∼ c ⇒ a ∼ c that is closed under translation a ∼ b ⇒ a + c ∼ b + c

Christopher O’Neill (SDSU) Computing delta sets of affine semigroups Feb 10, 2020 19 / 27

slide-125
SLIDE 125

Commutative algebra hiding in the background

Fix an affine semigroup S = n1, . . . , nk ⊂ Zd

≥0.

n = a1n1 + · · · + aknk

  • a = (a1, . . . , ak) ∈ Zk

≥0

Factorization homomorphism: π : Zk

≥0

− → n1, . . . , nk a − → a1n1 + · · · + aknk Monomial map:

Definition

The kernel ker π is the relation ∼ on Zk

≥0 with a ∼ b whenever

π(a) = π(b) ker π is a congruence: an equivalence relation a ∼ a a ∼ b ⇒ b ∼ a a ∼ b and b ∼ c ⇒ a ∼ c that is closed under translation a ∼ b ⇒ a + c ∼ b + c

Christopher O’Neill (SDSU) Computing delta sets of affine semigroups Feb 10, 2020 19 / 27

slide-126
SLIDE 126

Commutative algebra hiding in the background

Fix an affine semigroup S = n1, . . . , nk ⊂ Zd

≥0.

n = a1n1 + · · · + aknk

  • a = (a1, . . . , ak) ∈ Zk

≥0

Factorization homomorphism: π : Zk

≥0

− → n1, . . . , nk a − → a1n1 + · · · + aknk Monomial map: ϕ : k[x1, . . . , xk] − → k[w] xi − → wni

Definition

The kernel ker π is the relation ∼ on Zk

≥0 with a ∼ b whenever

π(a) = π(b) ker π is a congruence: an equivalence relation a ∼ a a ∼ b ⇒ b ∼ a a ∼ b and b ∼ c ⇒ a ∼ c that is closed under translation a ∼ b ⇒ a + c ∼ b + c

Christopher O’Neill (SDSU) Computing delta sets of affine semigroups Feb 10, 2020 19 / 27

slide-127
SLIDE 127

Commutative algebra hiding in the background

Fix an affine semigroup S = n1, . . . , nk ⊂ Zd

≥0.

n = a1n1 + · · · + aknk

  • a = (a1, . . . , ak) ∈ Zk

≥0

Factorization homomorphism: π : Zk

≥0

− → n1, . . . , nk a − → a1n1 + · · · + aknk Monomial map: ϕ : k[x1, . . . , xk] − → k[w] xi − → wni

Definition

The kernel ker π is the relation ∼ on Zk

≥0 with a ∼ b whenever

π(a) = π(b) xa − xb ∈ IS = ker ϕ ker π is a congruence: an equivalence relation a ∼ a a ∼ b ⇒ b ∼ a a ∼ b and b ∼ c ⇒ a ∼ c that is closed under translation a ∼ b ⇒ a + c ∼ b + c

Christopher O’Neill (SDSU) Computing delta sets of affine semigroups Feb 10, 2020 19 / 27

slide-128
SLIDE 128

Commutative algebra hiding in the background

Fix an affine semigroup S = n1, . . . , nk ⊂ Zd

≥0.

n = a1n1 + · · · + aknk

  • a = (a1, . . . , ak) ∈ Zk

≥0

Factorization homomorphism: π : Zk

≥0

− → n1, . . . , nk a − → a1n1 + · · · + aknk Monomial map: ϕ : k[x1, . . . , xk] − → k[w] xi − → wni

Definition

The kernel ker π is the relation ∼ on Zk

≥0 with a ∼ b whenever

π(a) = π(b) xa − xb ∈ IS = ker ϕ ker π is a congruence: an equivalence relation a ∼ a a ∼ b ⇒ b ∼ a a ∼ b and b ∼ c ⇒ a ∼ c xa − xa = 0 ∈ IS that is closed under translation a ∼ b ⇒ a + c ∼ b + c

Christopher O’Neill (SDSU) Computing delta sets of affine semigroups Feb 10, 2020 19 / 27

slide-129
SLIDE 129

Commutative algebra hiding in the background

Fix an affine semigroup S = n1, . . . , nk ⊂ Zd

≥0.

n = a1n1 + · · · + aknk

  • a = (a1, . . . , ak) ∈ Zk

≥0

Factorization homomorphism: π : Zk

≥0

− → n1, . . . , nk a − → a1n1 + · · · + aknk Monomial map: ϕ : k[x1, . . . , xk] − → k[w] xi − → wni

Definition

The kernel ker π is the relation ∼ on Zk

≥0 with a ∼ b whenever

π(a) = π(b) xa − xb ∈ IS = ker ϕ ker π is a congruence: an equivalence relation a ∼ a a ∼ b ⇒ b ∼ a a ∼ b and b ∼ c ⇒ a ∼ c xa − xa = 0 ∈ IS xa − xb ∈ IS ⇒ xb − xa ∈ IS that is closed under translation a ∼ b ⇒ a + c ∼ b + c

Christopher O’Neill (SDSU) Computing delta sets of affine semigroups Feb 10, 2020 19 / 27

slide-130
SLIDE 130

Commutative algebra hiding in the background

Fix an affine semigroup S = n1, . . . , nk ⊂ Zd

≥0.

n = a1n1 + · · · + aknk

  • a = (a1, . . . , ak) ∈ Zk

≥0

Factorization homomorphism: π : Zk

≥0

− → n1, . . . , nk a − → a1n1 + · · · + aknk Monomial map: ϕ : k[x1, . . . , xk] − → k[w] xi − → wni

Definition

The kernel ker π is the relation ∼ on Zk

≥0 with a ∼ b whenever

π(a) = π(b) xa − xb ∈ IS = ker ϕ ker π is a congruence: an equivalence relation a ∼ a a ∼ b ⇒ b ∼ a a ∼ b and b ∼ c ⇒ a ∼ c xa − xa = 0 ∈ IS xa − xb ∈ IS ⇒ xb − xa ∈ IS (xa − xb) + (xb − xc) = xa − xc that is closed under translation a ∼ b ⇒ a + c ∼ b + c

Christopher O’Neill (SDSU) Computing delta sets of affine semigroups Feb 10, 2020 19 / 27

slide-131
SLIDE 131

Commutative algebra hiding in the background

Fix an affine semigroup S = n1, . . . , nk ⊂ Zd

≥0.

n = a1n1 + · · · + aknk

  • a = (a1, . . . , ak) ∈ Zk

≥0

Factorization homomorphism: π : Zk

≥0

− → n1, . . . , nk a − → a1n1 + · · · + aknk Monomial map: ϕ : k[x1, . . . , xk] − → k[w] xi − → wni

Definition

The kernel ker π is the relation ∼ on Zk

≥0 with a ∼ b whenever

π(a) = π(b) xa − xb ∈ IS = ker ϕ ker π is a congruence: an equivalence relation a ∼ a a ∼ b ⇒ b ∼ a a ∼ b and b ∼ c ⇒ a ∼ c xa − xa = 0 ∈ IS xa − xb ∈ IS ⇒ xb − xa ∈ IS (xa − xb) + (xb − xc) = xa − xc that is closed under translation a ∼ b ⇒ a + c ∼ b + c xa − xb ∈ IS ⇒ xc(xa − xb) ∈ IS

Christopher O’Neill (SDSU) Computing delta sets of affine semigroups Feb 10, 2020 19 / 27

slide-132
SLIDE 132

Commutative algebra hiding in the background

Fix an affine semigroup S = n1, . . . , nk ⊂ Zd

≥0.

n = a1n1 + · · · + aknk

  • a = (a1, . . . , ak) ∈ Zk

≥0

Factorization homomorphism: π : Zk

≥0

− → n1, . . . , nk a − → a1n1 + · · · + aknk Monomial map: ϕ : k[x1, . . . , xk] − → k[w] xi − → wni

Example

S = 6, 9, 20:

Christopher O’Neill (SDSU) Computing delta sets of affine semigroups Feb 10, 2020 20 / 27

slide-133
SLIDE 133

Commutative algebra hiding in the background

Fix an affine semigroup S = n1, . . . , nk ⊂ Zd

≥0.

n = a1n1 + · · · + aknk

  • a = (a1, . . . , ak) ∈ Zk

≥0

Factorization homomorphism: π : Zk

≥0

− → n1, . . . , nk a − → a1n1 + · · · + aknk Monomial map: ϕ : k[x1, . . . , xk] − → k[w] xi − → wni

Example

S = 6, 9, 20: IS = x3 − y2, x4y4 − z3 ⊂ k[x, y, z]

Christopher O’Neill (SDSU) Computing delta sets of affine semigroups Feb 10, 2020 20 / 27

slide-134
SLIDE 134

Commutative algebra hiding in the background

Fix an affine semigroup S = n1, . . . , nk ⊂ Zd

≥0.

n = a1n1 + · · · + aknk

  • a = (a1, . . . , ak) ∈ Zk

≥0

Factorization homomorphism: π : Zk

≥0

− → n1, . . . , nk a − → a1n1 + · · · + aknk Monomial map: ϕ : k[x1, . . . , xk] − → k[w] xi − → wni

Example

S = 6, 9, 20: IS = x3 − y2, x4y4 − z3 ⊂ k[x, y, z] Z(18):

Christopher O’Neill (SDSU) Computing delta sets of affine semigroups Feb 10, 2020 20 / 27

slide-135
SLIDE 135

Commutative algebra hiding in the background

Fix an affine semigroup S = n1, . . . , nk ⊂ Zd

≥0.

n = a1n1 + · · · + aknk

  • a = (a1, . . . , ak) ∈ Zk

≥0

Factorization homomorphism: π : Zk

≥0

− → n1, . . . , nk a − → a1n1 + · · · + aknk Monomial map: ϕ : k[x1, . . . , xk] − → k[w] xi − → wni

Example

S = 6, 9, 20: IS = x3 − y2, x4y4 − z3 ⊂ k[x, y, z] Z(18): Z(60):

Christopher O’Neill (SDSU) Computing delta sets of affine semigroups Feb 10, 2020 20 / 27

slide-136
SLIDE 136

Commutative algebra hiding in the background

Fix an affine semigroup S = n1, . . . , nk ⊂ Zd

≥0.

n = a1n1 + · · · + aknk

  • a = (a1, . . . , ak) ∈ Zk

≥0

Factorization homomorphism: π : Zk

≥0

− → n1, . . . , nk a − → a1n1 + · · · + aknk Monomial map: ϕ : k[x1, . . . , xk] − → k[w] xi − → wni

Example

S = 6, 9, 20: IS = x3 − y2, x4y4 − z3 ⊂ k[x, y, z] Z(18): Z(60):

x7y 2 − x4y 4 = x4y 2(x3 − y 2)

Christopher O’Neill (SDSU) Computing delta sets of affine semigroups Feb 10, 2020 20 / 27

slide-137
SLIDE 137

Commutative algebra hiding in the background

Fix an affine semigroup S = n1, . . . , nk ⊂ Zd

≥0.

n = a1n1 + · · · + aknk

  • a = (a1, . . . , ak) ∈ Zk

≥0

Factorization homomorphism: π : Zk

≥0

− → n1, . . . , nk a − → a1n1 + · · · + aknk Monomial map: ϕ : k[x1, . . . , xk] − → k[w] xi − → wni

Example

S = 6, 9, 20: IS = x3 − y2, x4y4 − z3 ⊂ k[x, y, z] Z(18): Z(60):

x7y 2 − x4y 4 = x4y 2(x3 − y 2) x7y 2 − z3 = (x7y 2 − x4y 4) + (x4y 4 − z3)

Christopher O’Neill (SDSU) Computing delta sets of affine semigroups Feb 10, 2020 20 / 27

slide-138
SLIDE 138

Commutative algebra hiding in the background

Fix an affine semigroup S = n1, . . . , nk ⊂ Zd

≥0.

n = a1n1 + · · · + aknk

  • a = (a1, . . . , ak) ∈ Zk

≥0

Factorization homomorphism: π : Zk

≥0

− → n1, . . . , nk a − → a1n1 + · · · + aknk Monomial map: ϕ : k[x1, . . . , xk] − → k[w] xi − → wni

Example

S = 6, 9, 20: IS = x3 − y2, x4y4 − z3 ⊂ k[x, y, z] Z(18): Z(60):

x7y 2 − x4y 4 = x4y 2(x3 − y 2) x7y 2 − z3 = (x7y 2 − x4y 4) + (x4y 4 − z3) Generating ⇔ π−1(n) connected set for IS for all n ∈ S

Christopher O’Neill (SDSU) Computing delta sets of affine semigroups Feb 10, 2020 20 / 27

slide-139
SLIDE 139

Commutative algebra hiding in the background

Fix an affine semigroup S = n1, . . . , nk ⊂ Zd

≥0.

n = a1n1 + · · · + aknk

  • a = (a1, . . . , ak) ∈ Zk

≥0

Factorization homomorphism: π : Zk

≥0

− → n1, . . . , nk a − → a1n1 + · · · + aknk Monomial map: ϕ : k[x1, . . . , xk] − → k[w] xi − → wni

Example

S = 6, 9, 20: IS = x3 − y2, x4y4 − z3 ⊂ k[x, y, z] Z(18): Z(60):

Christopher O’Neill (SDSU) Computing delta sets of affine semigroups Feb 10, 2020 21 / 27

slide-140
SLIDE 140

Commutative algebra hiding in the background

Fix an affine semigroup S = n1, . . . , nk ⊂ Zd

≥0.

n = a1n1 + · · · + aknk

  • a = (a1, . . . , ak) ∈ Zk

≥0

Factorization homomorphism: π : Zk

≥0

− → n1, . . . , nk a − → a1n1 + · · · + aknk Monomial map: ϕ : k[x1, . . . , xk] − → k[w] xi − → wni

Example

S = 6, 9, 20: IS = x3 − y2, x4y4 − z3 ⊂ k[x, y, z] Z(18): Z(60): All minimal generating sets:

Christopher O’Neill (SDSU) Computing delta sets of affine semigroups Feb 10, 2020 21 / 27

slide-141
SLIDE 141

Commutative algebra hiding in the background

Fix an affine semigroup S = n1, . . . , nk ⊂ Zd

≥0.

n = a1n1 + · · · + aknk

  • a = (a1, . . . , ak) ∈ Zk

≥0

Factorization homomorphism: π : Zk

≥0

− → n1, . . . , nk a − → a1n1 + · · · + aknk Monomial map: ϕ : k[x1, . . . , xk] − → k[w] xi − → wni

Example

S = 6, 9, 20: IS = x3 − y2, x4y4 − z3 ⊂ k[x, y, z] Z(18): Z(60): All minimal generating sets: IS = x3 − y2, x10 − z3 = x3 − y2, x7y2 − z3 = x3 − y2, x4y4 − z3 = x3 − y2, x6y − z3

Christopher O’Neill (SDSU) Computing delta sets of affine semigroups Feb 10, 2020 21 / 27

slide-142
SLIDE 142

Commutative algebra hiding in the background

Fix an affine semigroup S = n1, . . . , nk ⊂ Zd

≥0.

n = a1n1 + · · · + aknk

  • a = (a1, . . . , ak) ∈ Zk

≥0

Factorization homomorphism: π : Zk

≥0

− → n1, . . . , nk a − → a1n1 + · · · + aknk Monomial map: ϕ : k[x1, . . . , xk] − → k[w] xi − → wni

Example

S = 6, 9, 20: IS = x3 − y2, x4y4 − z3 ⊂ k[x, y, z] Z(18): Z(60): All minimal generating sets: IS = x3 − y2, x10 − z3 = x3 − y2, x7y2 − z3 = x3 − y2, x4y4 − z3 = x3 − y2, x6y − z3

Christopher O’Neill (SDSU) Computing delta sets of affine semigroups Feb 10, 2020 21 / 27

slide-143
SLIDE 143

Commutative algebra hiding in the background

Fix an affine semigroup S = n1, . . . , nk ⊂ Zd

≥0.

n = a1n1 + · · · + aknk

  • a = (a1, . . . , ak) ∈ Zk

≥0

Factorization homomorphism: π : Zk

≥0

− → n1, . . . , nk a − → a1n1 + · · · + aknk Monomial map: ϕ : k[x1, . . . , xk] − → k[w] xi − → wni

Example

S = 6, 9, 20: IS = x3 − y2, x4y4 − z3 ⊂ k[x, y, z] Z(18): Z(60): All minimal generating sets: IS = x3 − y2, x10 − z3 = x3 − y2, x7y2 − z3 = x3 − y2, x4y4 − z3 = x3 − y2, x6y − z3

Christopher O’Neill (SDSU) Computing delta sets of affine semigroups Feb 10, 2020 21 / 27

slide-144
SLIDE 144

Commutative algebra hiding in the background

Fix an affine semigroup S = n1, . . . , nk ⊂ Zd

≥0.

n = a1n1 + · · · + aknk

  • a = (a1, . . . , ak) ∈ Zk

≥0

Factorization homomorphism: π : Zk

≥0

− → n1, . . . , nk a − → a1n1 + · · · + aknk Monomial map: ϕ : k[x1, . . . , xk] − → k[w] xi − → wni

Example

S = 6, 9, 20: IS = x3 − y2, x4y4 − z3 ⊂ k[x, y, z] Z(18): Z(60): All minimal generating sets: IS = x3 − y2, x10 − z3 = x3 − y2, x7y2 − z3 = x3 − y2, x4y4 − z3 = x3 − y2, x6y − z3

Christopher O’Neill (SDSU) Computing delta sets of affine semigroups Feb 10, 2020 21 / 27

slide-145
SLIDE 145

Commutative algebra hiding in the background

Fix an affine semigroup S = n1, . . . , nk ⊂ Zd

≥0.

n = a1n1 + · · · + aknk

  • a = (a1, . . . , ak) ∈ Zk

≥0

Factorization homomorphism: π : Zk

≥0

− → n1, . . . , nk a − → a1n1 + · · · + aknk Monomial map: ϕ : k[x1, . . . , xk] − → k[w] xi − → wni

Example

S = 6, 9, 20: IS = x3 − y2, x4y4 − z3 ⊂ k[x, y, z] Z(18): Z(60): All minimal generating sets: IS = x3 − y2, x10 − z3 = x3 − y2, x7y2 − z3 = x3 − y2, x4y4 − z3 = x3 − y2, x6y − z3

Christopher O’Neill (SDSU) Computing delta sets of affine semigroups Feb 10, 2020 21 / 27

slide-146
SLIDE 146

Commutative algebra hiding in the background

Fix an affine semigroup S = n1, . . . , nk ⊂ Zd

≥0.

n = a1n1 + · · · + aknk

  • a = (a1, . . . , ak) ∈ Zk

≥0

Factorization homomorphism: π : Zk

≥0

− → n1, . . . , nk a − → a1n1 + · · · + aknk Monomial map: ϕ : k[x1, . . . , xk] − → k[w] xi − → wni

Example

S = 6, 9, 20: IS = x3 − y2, x4y4 − z3 ⊂ k[x, y, z] Z(18): Z(60): All minimal generating sets: IS = x3 − y2, x10 − z3 = x3 − y2, x7y2 − z3 = x3 − y2, x4y4 − z3 = x3 − y2, x6y − z3

Christopher O’Neill (SDSU) Computing delta sets of affine semigroups Feb 10, 2020 21 / 27

slide-147
SLIDE 147

Commutative algebra hiding in the background

S = n1, . . . , nk ⊂ Zd

≥0 (affine)

π : Zk

≥0

− → S

Christopher O’Neill (SDSU) Computing delta sets of affine semigroups Feb 10, 2020 22 / 27

slide-148
SLIDE 148

Commutative algebra hiding in the background

S = n1, . . . , nk ⊂ Zd

≥0 (affine)

π : Zk

≥0

− → S A larger example: S = 13, 44, 106, 120.

Christopher O’Neill (SDSU) Computing delta sets of affine semigroups Feb 10, 2020 22 / 27

slide-149
SLIDE 149

Commutative algebra hiding in the background

S = n1, . . . , nk ⊂ Zd

≥0 (affine)

π : Zk

≥0

− → S A larger example: S = 13, 44, 106, 120. IS = x6

1 x2 4 − x3 3, x2 1 x3 − x3 2, x14 1 x2 − x3 x4, x16 1 − x2 2 x4, x6 1 x4 2 x3 − x3 4

Christopher O’Neill (SDSU) Computing delta sets of affine semigroups Feb 10, 2020 22 / 27

slide-150
SLIDE 150

Commutative algebra hiding in the background

S = n1, . . . , nk ⊂ Zd

≥0 (affine)

π : Zk

≥0

− → S A larger example: S = 13, 44, 106, 120. IS = x6

1 x2 4 − x3 3, x2 1 x3 − x3 2, x14 1 x2 − x3 x4, x16 1 − x2 2 x4, x6 1 x4 2 x3 − x3 4

< l a t e x i t s h a 1 _ b a s e 6 4 = " N U W e E n b E i V j O y W k A N B n m W 9 / g v x I = " > A A A B 9 X i c b V D L S s N A F L 2 p r 1 p f U Z d u B
  • t
Q N y W p g l W 3 L i s Y B / Y x j K Z T t q h k m Y m S g l 9 D / c u F D E r f / i z r 9 x m m a h r Q c G D u f c y z 1 z / J g z p R 3 n 2 y q s r W 9 s b h W 3 S z u 7 e / s H 9 u F R W W J J L R F I h 7 J r
  • 8
V 5 U z Q l m a a 2 4 s K Q 5 9 T j v + 5 H r u d x 6 p V C w S d 3
  • a
U y / E I 8 E C R r A 2 k M / x H q s A n R f c S 9 q 5 w O 7 7 F S d D G i V u D k p Q 4 7 m w P 7 q D y O S h F R
  • w
r F S P d e J t Z d i q R n h d F b q J 4 r G m E z w i P Y M F T i k y k u z 1 D N Z p Q h C i J p n t A
  • U
3 9 v p D h U a h r 6 Z j J L u e z N x f + 8 X q K D u p c y E S e a C r I 4 F C Q c 6 Q j N K B D J i n R f G
  • I
J p K Z r I i M s c R E m 6 J K p g R 3 + c u r p F 2 r u k 7 V v b s N + p 5 H U U 4 g V O
  • g
A t X I A b a E I L C E h 4 h l d 4 s 5 6 s F + v d + l i M F q x 8 5 x j + w P r 8 A Z / R k T w = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " c v x g x t x a b S p O H M j q Z h 6 h B M h M r n I = " > A A A B 9 X i c b V D L S g M x F L 3 x W e u r 6 t J N s A h 1 U 2 Z U s M u C G 5 c V 7 A P b s W T S T B u a y Q x J R i l D / 8 O N C X c + i / u / B v T 6 S y 9 U D g c M 6 9 3 J P j x 4 J r 4 z j f a G V 1 b X 1 j s 7 B V 3 N 7 Z 3 d s v H R y 2 d J Q
  • y
p
  • E
p H q + E Q z w S V r G m 4 E 6 8 S K k d A X r O 2 P r 2 d + + 5 E p z S N 5 Z y Y x 8 I y l D z g l B g r P f R C Y k Y 6 w P e V C 7 d 2 1 i + V n a q T A S 8 T N y d l y N H
  • l
7 5 6 g 4 g m I Z O G C q J 1 1 3 V i 4 6 V E G U 4 F m x Z 7 i W Y x
  • W
M y Z F 1 L J Q m Z 9 t I s 9 R S f W m W A g j Z J w 3 O 1 N 8 b K Q m 1 n
  • S
+ n c x S L n
  • z
8 T + v m 5 i g 5 q V c x
  • l
h k s 4 P B Y n A J s K z C v C A K a N m F h C q O I 2 K 6 Y j
  • g
g 1 t q i i L c F d / P I y a Z 1 X X a f q 3 l 6 W 6 7 W 8 j g I c w w l U w I U r q M M N N K A J F B Q 8 w y u 8
  • S
f g t 7 R x 3 x B e U 7 R / A H 6 P M H q P G R Q g = = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " g V H x Y 3 M W F v x P c y e t l F u C P s Q B z k = " > A A A B 9 X i c b V D L S s N A F L 2 p r 1 p f U Z d u B
  • t
Q N y U p
  • l
W 3 L i s Y B / Y x j K Z T t q h k m Y m S g l 9 D / c u F D E r f / i z r 9 x m m a h r Q c G D u f c y z 1 z / J g z p R 3 n 2 y q s r W 9 s b h W 3 S z u 7 e / s H 9 u F R W W J J L R F I h 7 J r
  • 8
V 5 U z Q l m a a 2 4 s K Q 5 9 T j v + 5 H r u d x 6 p V C w S d 3
  • a
U y / E I 8 E C R r A 2 k M / x H q s A n R f q d U u z w d 2 2 a k 6 G d A q c X N S h h z N g f 3 V H Y k C a n Q h G O l e q 4 T a y / F U j P C 6 a z U T x S N M Z n g E e Z K n B I l Z d m q W f
  • z
C h D F E T S P K F R p v 7 e S H G
  • 1
D T z W S W c t m b i / 9 5 v U Q H d S 9 l I k 4 F W R x K E g 4 h G a V 4 C G T F K i + d Q Q T C Q z W R E Z Y 4 m J N k W V T A n u 8 p d X S b t W d Z 2 q e 3 t R b t T z O
  • p
w A q d Q A R e u
  • A
E 3 I Q W E J D w D K / w Z j 1 Z L 9 a 7 9 b E Y L V j 5 z j H 8 g f X 5 A 6 X m k U A = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " q q n s r D D a 3 i H J Y B x U w X + L P Q F k 4 I = " > A A A B 9 X i c b V D L S g M x F L 3 x W e u r 6 t J N s A h 1 U 2 a K Y J c F N y 4 r 2 A e 2 Y 8 m k m T Y k x m S j F K G /
  • c
b F 4 q 4 9 V / c + T e m 1 l
  • 6
4 H A 4 Z x 7 u S f H j w X X x n G + d r 6 x u b W d m G n u L u 3 f 3 B Y O j p u 6 y h R l L V
  • J
C L V 9 Y l m g k v W M t w I 1
  • V
I 6 E v W M e f X M / 9 z i N T m k f y z k x j 5
  • V
k J H n A K T F W e u i H x I x 1 g O 8 r N a d + M S i V n a q T A a 8 S N y d l y N E c l L 7 6 w 4 g m I Z O G C q J 1 z 3 V i 4 6 V E G U 4 F m x X 7 i W Y x
  • R
M y Y j 1 L J Q m Z 9 t I s 9 Q y f W 2 W I g j Z J w 3 O 1 N 8 b K Q m 1 n
  • a
+ n c x S L n t z 8 T + v l 5 i g 7 q V c x
  • l
h k i 4 O B Y n A J s L z C v C Q K a N m F p C q O I 2 K 6 Z j
  • g
g 1 t q i i L c F d / v I q a d e q r l N 1 b y / L j X p e R w F O 4 Q w q 4 M I V N O A G m t A C C g q e 4 R X e B N 6 Q e /
  • Y
z G 6 h v K d E / g D 9 P k D p e S R Q A = = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " G s f Q v I E A V f q R A G p W v W U 6 Q G i N 8 = " > A A A B 9 X i c b V B N T 8 J A F H z F L 8 Q v 1 K O X j c Q E L 6 R V
  • x
x J v H j E x A I R K t k u W 9 i w 3 T a 7 W w 1 p + B 9 e P G i M V / + L N / + N S + l B w U k 2 m c y 8 l z c 7 f s y Z r b 9 b R V W V t f W N 4 q b p a 3 t n d 2 9 8 v 5 B S W J J N Q l E Y 9 k x 8 e K c i a
  • q
5 n m t B N L i k O f 7 Y / v p 7 5 7 U c q F Y v E n Z 7 E 1 A v x U L C A E a y N 9 N A L s R 6 p A N 1 X z y / t 3 6 5 Y t f s D G i Z O D m p Q I 5 m v / z V G Q k C a n Q h G O l u
  • 4
d a y / F U j P C 6 b T U S x S N M R n j I e a K n B I l Z d m q a f
  • x
C g D F E T S P K F R p v 7 e S H G
  • 1
C T z W S W c t G b i f 9 5 3 U Q H d S 9 l I k 4 F W R + K E g 4 h G a V Y A G T F K i + c Q Q T C Q z W R E Z Y Y m J N k W V T A n O 4 p e X S e u s 5 t g 1 5 / a i q j n d R T h C I 6 h C g 5 c Q Q N u
  • A
k u E J D w D K / w Z j 1 Z L 9 a 7 9 T E f L V j 5 z i H 8 g f X 5 A 6 R n k T 8 = < / l a t e x i t >

Christopher O’Neill (SDSU) Computing delta sets of affine semigroups Feb 10, 2020 22 / 27

slide-151
SLIDE 151

Commutative algebra hiding in the background

S = n1, . . . , nk ⊂ Zd

≥0 (affine)

π : Zk

≥0

− → S A larger example: S = 13, 44, 106, 120. IS = x6

1 x2 4 − x3 3, x2 1 x3 − x3 2, x14 1 x2 − x3 x4, x16 1 − x2 2 x4, x6 1 x4 2 x3 − x3 4

< l a t e x i t s h a 1 _ b a s e 6 4 = " N U W e E n b E i V j O y W k A N B n m W 9 / g v x I = " > A A A B 9 X i c b V D L S s N A F L 2 p r 1 p f U Z d u B
  • t
Q N y W p g l W 3 L i s Y B / Y x j K Z T t q h k m Y m S g l 9 D / c u F D E r f / i z r 9 x m m a h r Q c G D u f c y z 1 z / J g z p R 3 n 2 y q s r W 9 s b h W 3 S z u 7 e / s H 9 u F R W W J J L R F I h 7 J r
  • 8
V 5 U z Q l m a a 2 4 s K Q 5 9 T j v + 5 H r u d x 6 p V C w S d 3
  • a
U y / E I 8 E C R r A 2 k M / x H q s A n R f c S 9 q 5 w O 7 7 F S d D G i V u D k p Q 4 7 m w P 7 q D y O S h F R
  • w
r F S P d e J t Z d i q R n h d F b q J 4 r G m E z w i P Y M F T i k y k u z 1 D N Z p Q h C i J p n t A
  • U
3 9 v p D h U a h r 6 Z j J L u e z N x f + 8 X q K D u p c y E S e a C r I 4 F C Q c 6 Q j N K B D J i n R f G
  • I
J p K Z r I i M s c R E m 6 J K p g R 3 + c u r p F 2 r u k 7 V v b s N + p 5 H U U 4 g V O
  • g
A t X I A b a E I L C E h 4 h l d 4 s 5 6 s F + v d + l i M F q x 8 5 x j + w P r 8 A Z / R k T w = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " c v x g x t x a b S p O H M j q Z h 6 h B M h M r n I = " > A A A B 9 X i c b V D L S g M x F L 3 x W e u r 6 t J N s A h 1 U 2 Z U s M u C G 5 c V 7 A P b s W T S T B u a y Q x J R i l D / 8 O N C X c + i / u / B v T 6 S y 9 U D g c M 6 9 3 J P j x 4 J r 4 z j f a G V 1 b X 1 j s 7 B V 3 N 7 Z 3 d s v H R y 2 d J Q
  • y
p
  • E
p H q + E Q z w S V r G m 4 E 6 8 S K k d A X r O 2 P r 2 d + + 5 E p z S N 5 Z y Y x 8 I y l D z g l B g r P f R C Y k Y 6 w P e V C 7 d 2 1 i + V n a q T A S 8 T N y d l y N H
  • l
7 5 6 g 4 g m I Z O G C q J 1 1 3 V i 4 6 V E G U 4 F m x Z 7 i W Y x
  • W
M y Z F 1 L J Q m Z 9 t I s 9 R S f W m W A g j Z J w 3 O 1 N 8 b K Q m 1 n
  • S
+ n c x S L n
  • z
8 T + v m 5 i g 5 q V c x
  • l
h k s 4 P B Y n A J s K z C v C A K a N m F h C q O I 2 K 6 Y j
  • g
g 1 t q i i L c F d / P I y a Z 1 X X a f q 3 l 6 W 6 7 W 8 j g I c w w l U w I U r q M M N N K A J F B Q 8 w y u 8
  • S
f g t 7 R x 3 x B e U 7 R / A H 6 P M H q P G R Q g = = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " g V H x Y 3 M W F v x P c y e t l F u C P s Q B z k = " > A A A B 9 X i c b V D L S s N A F L 2 p r 1 p f U Z d u B
  • t
Q N y U p
  • l
W 3 L i s Y B / Y x j K Z T t q h k m Y m S g l 9 D / c u F D E r f / i z r 9 x m m a h r Q c G D u f c y z 1 z / J g z p R 3 n 2 y q s r W 9 s b h W 3 S z u 7 e / s H 9 u F R W W J J L R F I h 7 J r
  • 8
V 5 U z Q l m a a 2 4 s K Q 5 9 T j v + 5 H r u d x 6 p V C w S d 3
  • a
U y / E I 8 E C R r A 2 k M / x H q s A n R f q d U u z w d 2 2 a k 6 G d A q c X N S h h z N g f 3 V H Y k C a n Q h G O l e q 4 T a y / F U j P C 6 a z U T x S N M Z n g E e Z K n B I l Z d m q W f
  • z
C h D F E T S P K F R p v 7 e S H G
  • 1
D T z W S W c t m b i / 9 5 v U Q H d S 9 l I k 4 F W R x K E g 4 h G a V 4 C G T F K i + d Q Q T C Q z W R E Z Y 4 m J N k W V T A n u 8 p d X S b t W d Z 2 q e 3 t R b t T z O
  • p
w A q d Q A R e u
  • A
E 3 I Q W E J D w D K / w Z j 1 Z L 9 a 7 9 b E Y L V j 5 z j H 8 g f X 5 A 6 X m k U A = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " q q n s r D D a 3 i H J Y B x U w X + L P Q F k 4 I = " > A A A B 9 X i c b V D L S g M x F L 3 x W e u r 6 t J N s A h 1 U 2 a K Y J c F N y 4 r 2 A e 2 Y 8 m k m T Y k x m S j F K G /
  • c
b F 4 q 4 9 V / c + T e m 1 l
  • 6
4 H A 4 Z x 7 u S f H j w X X x n G + d r 6 x u b W d m G n u L u 3 f 3 B Y O j p u 6 y h R l L V
  • J
C L V 9 Y l m g k v W M t w I 1
  • V
I 6 E v W M e f X M / 9 z i N T m k f y z k x j 5
  • V
k J H n A K T F W e u i H x I x 1 g O 8 r N a d + M S i V n a q T A a 8 S N y d l y N E c l L 7 6 w 4 g m I Z O G C q J 1 z 3 V i 4 6 V E G U 4 F m x X 7 i W Y x
  • R
M y Y j 1 L J Q m Z 9 t I s 9 Q y f W 2 W I g j Z J w 3 O 1 N 8 b K Q m 1 n
  • a
+ n c x S L n t z 8 T + v l 5 i g 7 q V c x
  • l
h k i 4 O B Y n A J s L z C v C Q K a N m F p C q O I 2 K 6 Z j
  • g
g 1 t q i i L c F d / v I q a d e q r l N 1 b y / L j X p e R w F O 4 Q w q 4 M I V N O A G m t A C C g q e 4 R X e B N 6 Q e /
  • Y
z G 6 h v K d E / g D 9 P k D p e S R Q A = = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " G s f Q v I E A V f q R A G p W v W U 6 Q G i N 8 = " > A A A B 9 X i c b V B N T 8 J A F H z F L 8 Q v 1 K O X j c Q E L 6 R V
  • x
x J v H j E x A I R K t k u W 9 i w 3 T a 7 W w 1 p + B 9 e P G i M V / + L N / + N S + l B w U k 2 m c y 8 l z c 7 f s y Z r b 9 b R V W V t f W N 4 q b p a 3 t n d 2 9 8 v 5 B S W J J N Q l E Y 9 k x 8 e K c i a
  • q
5 n m t B N L i k O f 7 Y / v p 7 5 7 U c q F Y v E n Z 7 E 1 A v x U L C A E a y N 9 N A L s R 6 p A N 1 X z y / t 3 6 5 Y t f s D G i Z O D m p Q I 5 m v / z V G Q k C a n Q h G O l u
  • 4
d a y / F U j P C 6 b T U S x S N M R n j I e a K n B I l Z d m q a f
  • x
C g D F E T S P K F R p v 7 e S H G
  • 1
C T z W S W c t G b i f 9 5 3 U Q H d S 9 l I k 4 F W R + K E g 4 h G a V Y A G T F K i + c Q Q T C Q z W R E Z Y Y m J N k W V T A n O 4 p e X S e u s 5 t g 1 5 / a i q j n d R T h C I 6 h C g 5 c Q Q N u
  • A
k u E J D w D K / w Z j 1 Z L 9 a 7 9 T E f L V j 5 z i H 8 g f X 5 A 6 R n k T 8 = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " 1 X g H 5 c j n
  • 4
y c k T X 2
  • b
9 / z e R v 8 I = " > A A A B 9 X i c b V D L S s N A F L 2 p r 1 p f V Z d u B
  • t
Q N y U R i 1 W 3 L i s Y B / Y x j K Z T t q h k m Y m S g l 9 D / c u F D E r f / i z r 9 x k m a h r Q c G D u f c y z 1 z v I g z p W 3 7 2 y q s r W 9 s b h W 3 S z u 7 e / s H 5 c O j j g p j S W i b h D y U P Q 8 r y p m g b c p 7 1 I U h x 4 n H a 9 6 X X q d x + p V C w U d 3
  • W
U T f A Y 8 F 8 R r A 2 s M g w H q i f H R f r d f t 8 2 G 5 Y t f s D G i V O D m p Q I 7 W s P w 1 G I U k D q j Q h G O l +
  • 4
d a T f B U j P C 6 b w i B W N M J n i M e b K n B A l Z t k q e f
  • z
C g j 5 I f S P K F R p v 7 e S H C g 1 C z w z G S W c t l L x f + 8 f q z 9 h p s w E c W a C r I 4 5 M c c 6 R C l F a A R k 5 R
  • P
j M E E 8 l M V k Q m W G K i T V E l U 4 K z / O V V r m
  • O
X b N u b 2 s N B t 5 H U U 4 g V O
  • g
g N X I Q b a E E b C E h 4 h l d 4 s 5 6 s F + v d + l i M F q x 8 5 x j + w P r 8 A a X v k U A = < / l a t e x i t >

Christopher O’Neill (SDSU) Computing delta sets of affine semigroups Feb 10, 2020 22 / 27

slide-152
SLIDE 152

Commutative algebra hiding in the background

S = n1, . . . , nk ⊂ Zd

≥0 (affine)

π : Zk

≥0

− → S A larger example: S = 13, 44, 106, 120. IS = x6

1 x2 4 − x3 3, x2 1 x3 − x3 2, x14 1 x2 − x3 x4, x16 1 − x2 2 x4, x6 1 x4 2 x3 − x3 4

< l a t e x i t s h a 1 _ b a s e 6 4 = " N U W e E n b E i V j O y W k A N B n m W 9 / g v x I = " > A A A B 9 X i c b V D L S s N A F L 2 p r 1 p f U Z d u B
  • t
Q N y W p g l W 3 L i s Y B / Y x j K Z T t q h k m Y m S g l 9 D / c u F D E r f / i z r 9 x m m a h r Q c G D u f c y z 1 z / J g z p R 3 n 2 y q s r W 9 s b h W 3 S z u 7 e / s H 9 u F R W W J J L R F I h 7 J r
  • 8
V 5 U z Q l m a a 2 4 s K Q 5 9 T j v + 5 H r u d x 6 p V C w S d 3
  • a
U y / E I 8 E C R r A 2 k M / x H q s A n R f c S 9 q 5 w O 7 7 F S d D G i V u D k p Q 4 7 m w P 7 q D y O S h F R
  • w
r F S P d e J t Z d i q R n h d F b q J 4 r G m E z w i P Y M F T i k y k u z 1 D N Z p Q h C i J p n t A
  • U
3 9 v p D h U a h r 6 Z j J L u e z N x f + 8 X q K D u p c y E S e a C r I 4 F C Q c 6 Q j N K B D J i n R f G
  • I
J p K Z r I i M s c R E m 6 J K p g R 3 + c u r p F 2 r u k 7 V v b s N + p 5 H U U 4 g V O
  • g
A t X I A b a E I L C E h 4 h l d 4 s 5 6 s F + v d + l i M F q x 8 5 x j + w P r 8 A Z / R k T w = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " c v x g x t x a b S p O H M j q Z h 6 h B M h M r n I = " > A A A B 9 X i c b V D L S g M x F L 3 x W e u r 6 t J N s A h 1 U 2 Z U s M u C G 5 c V 7 A P b s W T S T B u a y Q x J R i l D / 8 O N C X c + i / u / B v T 6 S y 9 U D g c M 6 9 3 J P j x 4 J r 4 z j f a G V 1 b X 1 j s 7 B V 3 N 7 Z 3 d s v H R y 2 d J Q
  • y
p
  • E
p H q + E Q z w S V r G m 4 E 6 8 S K k d A X r O 2 P r 2 d + + 5 E p z S N 5 Z y Y x 8 I y l D z g l B g r P f R C Y k Y 6 w P e V C 7 d 2 1 i + V n a q T A S 8 T N y d l y N H
  • l
7 5 6 g 4 g m I Z O G C q J 1 1 3 V i 4 6 V E G U 4 F m x Z 7 i W Y x
  • W
M y Z F 1 L J Q m Z 9 t I s 9 R S f W m W A g j Z J w 3 O 1 N 8 b K Q m 1 n
  • S
+ n c x S L n
  • z
8 T + v m 5 i g 5 q V c x
  • l
h k s 4 P B Y n A J s K z C v C A K a N m F h C q O I 2 K 6 Y j
  • g
g 1 t q i i L c F d / P I y a Z 1 X X a f q 3 l 6 W 6 7 W 8 j g I c w w l U w I U r q M M N N K A J F B Q 8 w y u 8
  • S
f g t 7 R x 3 x B e U 7 R / A H 6 P M H q P G R Q g = = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " g V H x Y 3 M W F v x P c y e t l F u C P s Q B z k = " > A A A B 9 X i c b V D L S s N A F L 2 p r 1 p f U Z d u B
  • t
Q N y U p
  • l
W 3 L i s Y B / Y x j K Z T t q h k m Y m S g l 9 D / c u F D E r f / i z r 9 x m m a h r Q c G D u f c y z 1 z / J g z p R 3 n 2 y q s r W 9 s b h W 3 S z u 7 e / s H 9 u F R W W J J L R F I h 7 J r
  • 8
V 5 U z Q l m a a 2 4 s K Q 5 9 T j v + 5 H r u d x 6 p V C w S d 3
  • a
U y / E I 8 E C R r A 2 k M / x H q s A n R f q d U u z w d 2 2 a k 6 G d A q c X N S h h z N g f 3 V H Y k C a n Q h G O l e q 4 T a y / F U j P C 6 a z U T x S N M Z n g E e Z K n B I l Z d m q W f
  • z
C h D F E T S P K F R p v 7 e S H G
  • 1
D T z W S W c t m b i / 9 5 v U Q H d S 9 l I k 4 F W R x K E g 4 h G a V 4 C G T F K i + d Q Q T C Q z W R E Z Y 4 m J N k W V T A n u 8 p d X S b t W d Z 2 q e 3 t R b t T z O
  • p
w A q d Q A R e u
  • A
E 3 I Q W E J D w D K / w Z j 1 Z L 9 a 7 9 b E Y L V j 5 z j H 8 g f X 5 A 6 X m k U A = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " q q n s r D D a 3 i H J Y B x U w X + L P Q F k 4 I = " > A A A B 9 X i c b V D L S g M x F L 3 x W e u r 6 t J N s A h 1 U 2 a K Y J c F N y 4 r 2 A e 2 Y 8 m k m T Y k x m S j F K G /
  • c
b F 4 q 4 9 V / c + T e m 1 l
  • 6
4 H A 4 Z x 7 u S f H j w X X x n G + d r 6 x u b W d m G n u L u 3 f 3 B Y O j p u 6 y h R l L V
  • J
C L V 9 Y l m g k v W M t w I 1
  • V
I 6 E v W M e f X M / 9 z i N T m k f y z k x j 5
  • V
k J H n A K T F W e u i H x I x 1 g O 8 r N a d + M S i V n a q T A a 8 S N y d l y N E c l L 7 6 w 4 g m I Z O G C q J 1 z 3 V i 4 6 V E G U 4 F m x X 7 i W Y x
  • R
M y Y j 1 L J Q m Z 9 t I s 9 Q y f W 2 W I g j Z J w 3 O 1 N 8 b K Q m 1 n
  • a
+ n c x S L n t z 8 T + v l 5 i g 7 q V c x
  • l
h k i 4 O B Y n A J s L z C v C Q K a N m F p C q O I 2 K 6 Z j
  • g
g 1 t q i i L c F d / v I q a d e q r l N 1 b y / L j X p e R w F O 4 Q w q 4 M I V N O A G m t A C C g q e 4 R X e B N 6 Q e /
  • Y
z G 6 h v K d E / g D 9 P k D p e S R Q A = = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " G s f Q v I E A V f q R A G p W v W U 6 Q G i N 8 = " > A A A B 9 X i c b V B N T 8 J A F H z F L 8 Q v 1 K O X j c Q E L 6 R V
  • x
x J v H j E x A I R K t k u W 9 i w 3 T a 7 W w 1 p + B 9 e P G i M V / + L N / + N S + l B w U k 2 m c y 8 l z c 7 f s y Z r b 9 b R V W V t f W N 4 q b p a 3 t n d 2 9 8 v 5 B S W J J N Q l E Y 9 k x 8 e K c i a
  • q
5 n m t B N L i k O f 7 Y / v p 7 5 7 U c q F Y v E n Z 7 E 1 A v x U L C A E a y N 9 N A L s R 6 p A N 1 X z y / t 3 6 5 Y t f s D G i Z O D m p Q I 5 m v / z V G Q k C a n Q h G O l u
  • 4
d a y / F U j P C 6 b T U S x S N M R n j I e a K n B I l Z d m q a f
  • x
C g D F E T S P K F R p v 7 e S H G
  • 1
C T z W S W c t G b i f 9 5 3 U Q H d S 9 l I k 4 F W R + K E g 4 h G a V Y A G T F K i + c Q Q T C Q z W R E Z Y Y m J N k W V T A n O 4 p e X S e u s 5 t g 1 5 / a i q j n d R T h C I 6 h C g 5 c Q Q N u
  • A
k u E J D w D K / w Z j 1 Z L 9 a 7 9 T E f L V j 5 z i H 8 g f X 5 A 6 R n k T 8 = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " 1 X g H 5 c j n
  • 4
y c k T X 2
  • b
9 / z e R v 8 I = " > A A A B 9 X i c b V D L S s N A F L 2 p r 1 p f V Z d u B
  • t
Q N y U R i 1 W 3 L i s Y B / Y x j K Z T t q h k m Y m S g l 9 D / c u F D E r f / i z r 9 x k m a h r Q c G D u f c y z 1 z v I g z p W 3 7 2 y q s r W 9 s b h W 3 S z u 7 e / s H 5 c O j j g p j S W i b h D y U P Q 8 r y p m g b c p 7 1 I U h x 4 n H a 9 6 X X q d x + p V C w U d 3
  • W
U T f A Y 8 F 8 R r A 2 s M g w H q i f H R f r d f t 8 2 G 5 Y t f s D G i V O D m p Q I 7 W s P w 1 G I U k D q j Q h G O l +
  • 4
d a T f B U j P C 6 b w i B W N M J n i M e b K n B A l Z t k q e f
  • z
C g j 5 I f S P K F R p v 7 e S H C g 1 C z w z G S W c t l L x f + 8 f q z 9 h p s w E c W a C r I 4 5 M c c 6 R C l F a A R k 5 R
  • P
j M E E 8 l M V k Q m W G K i T V E l U 4 K z / O V V r m
  • O
X b N u b 2 s N B t 5 H U U 4 g V O
  • g
g N X I Q b a E E b C E h 4 h l d 4 s 5 6 s F + v d + l i M F q x 8 5 x j + w P r 8 A a X v k U A = < / l a t e x i t >

Christopher O’Neill (SDSU) Computing delta sets of affine semigroups Feb 10, 2020 22 / 27

slide-153
SLIDE 153

Commutative algebra hiding in the background

S = n1, . . . , nk ⊂ Zd

≥0 (affine)

π : Zk

≥0

− → S A larger example: S = 13, 44, 106, 120. IS = x6

1 x2 4 − x3 3, x2 1 x3 − x3 2, x14 1 x2 − x3 x4, x16 1 − x2 2 x4, x6 1 x4 2 x3 − x3 4

< l a t e x i t s h a 1 _ b a s e 6 4 = " N U W e E n b E i V j O y W k A N B n m W 9 / g v x I = " > A A A B 9 X i c b V D L S s N A F L 2 p r 1 p f U Z d u B
  • t
Q N y W p g l W 3 L i s Y B / Y x j K Z T t q h k m Y m S g l 9 D / c u F D E r f / i z r 9 x m m a h r Q c G D u f c y z 1 z / J g z p R 3 n 2 y q s r W 9 s b h W 3 S z u 7 e / s H 9 u F R W W J J L R F I h 7 J r
  • 8
V 5 U z Q l m a a 2 4 s K Q 5 9 T j v + 5 H r u d x 6 p V C w S d 3
  • a
U y / E I 8 E C R r A 2 k M / x H q s A n R f c S 9 q 5 w O 7 7 F S d D G i V u D k p Q 4 7 m w P 7 q D y O S h F R
  • w
r F S P d e J t Z d i q R n h d F b q J 4 r G m E z w i P Y M F T i k y k u z 1 D N Z p Q h C i J p n t A
  • U
3 9 v p D h U a h r 6 Z j J L u e z N x f + 8 X q K D u p c y E S e a C r I 4 F C Q c 6 Q j N K B D J i n R f G
  • I
J p K Z r I i M s c R E m 6 J K p g R 3 + c u r p F 2 r u k 7 V v b s N + p 5 H U U 4 g V O
  • g
A t X I A b a E I L C E h 4 h l d 4 s 5 6 s F + v d + l i M F q x 8 5 x j + w P r 8 A Z / R k T w = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " c v x g x t x a b S p O H M j q Z h 6 h B M h M r n I = " > A A A B 9 X i c b V D L S g M x F L 3 x W e u r 6 t J N s A h 1 U 2 Z U s M u C G 5 c V 7 A P b s W T S T B u a y Q x J R i l D / 8 O N C X c + i / u / B v T 6 S y 9 U D g c M 6 9 3 J P j x 4 J r 4 z j f a G V 1 b X 1 j s 7 B V 3 N 7 Z 3 d s v H R y 2 d J Q
  • y
p
  • E
p H q + E Q z w S V r G m 4 E 6 8 S K k d A X r O 2 P r 2 d + + 5 E p z S N 5 Z y Y x 8 I y l D z g l B g r P f R C Y k Y 6 w P e V C 7 d 2 1 i + V n a q T A S 8 T N y d l y N H
  • l
7 5 6 g 4 g m I Z O G C q J 1 1 3 V i 4 6 V E G U 4 F m x Z 7 i W Y x
  • W
M y Z F 1 L J Q m Z 9 t I s 9 R S f W m W A g j Z J w 3 O 1 N 8 b K Q m 1 n
  • S
+ n c x S L n
  • z
8 T + v m 5 i g 5 q V c x
  • l
h k s 4 P B Y n A J s K z C v C A K a N m F h C q O I 2 K 6 Y j
  • g
g 1 t q i i L c F d / P I y a Z 1 X X a f q 3 l 6 W 6 7 W 8 j g I c w w l U w I U r q M M N N K A J F B Q 8 w y u 8
  • S
f g t 7 R x 3 x B e U 7 R / A H 6 P M H q P G R Q g = = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " g V H x Y 3 M W F v x P c y e t l F u C P s Q B z k = " > A A A B 9 X i c b V D L S s N A F L 2 p r 1 p f U Z d u B
  • t
Q N y U p
  • l
W 3 L i s Y B / Y x j K Z T t q h k m Y m S g l 9 D / c u F D E r f / i z r 9 x m m a h r Q c G D u f c y z 1 z / J g z p R 3 n 2 y q s r W 9 s b h W 3 S z u 7 e / s H 9 u F R W W J J L R F I h 7 J r
  • 8
V 5 U z Q l m a a 2 4 s K Q 5 9 T j v + 5 H r u d x 6 p V C w S d 3
  • a
U y / E I 8 E C R r A 2 k M / x H q s A n R f q d U u z w d 2 2 a k 6 G d A q c X N S h h z N g f 3 V H Y k C a n Q h G O l e q 4 T a y / F U j P C 6 a z U T x S N M Z n g E e Z K n B I l Z d m q W f
  • z
C h D F E T S P K F R p v 7 e S H G
  • 1
D T z W S W c t m b i / 9 5 v U Q H d S 9 l I k 4 F W R x K E g 4 h G a V 4 C G T F K i + d Q Q T C Q z W R E Z Y 4 m J N k W V T A n u 8 p d X S b t W d Z 2 q e 3 t R b t T z O
  • p
w A q d Q A R e u
  • A
E 3 I Q W E J D w D K / w Z j 1 Z L 9 a 7 9 b E Y L V j 5 z j H 8 g f X 5 A 6 X m k U A = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " q q n s r D D a 3 i H J Y B x U w X + L P Q F k 4 I = " > A A A B 9 X i c b V D L S g M x F L 3 x W e u r 6 t J N s A h 1 U 2 a K Y J c F N y 4 r 2 A e 2 Y 8 m k m T Y k x m S j F K G /
  • c
b F 4 q 4 9 V / c + T e m 1 l
  • 6
4 H A 4 Z x 7 u S f H j w X X x n G + d r 6 x u b W d m G n u L u 3 f 3 B Y O j p u 6 y h R l L V
  • J
C L V 9 Y l m g k v W M t w I 1
  • V
I 6 E v W M e f X M / 9 z i N T m k f y z k x j 5
  • V
k J H n A K T F W e u i H x I x 1 g O 8 r N a d + M S i V n a q T A a 8 S N y d l y N E c l L 7 6 w 4 g m I Z O G C q J 1 z 3 V i 4 6 V E G U 4 F m x X 7 i W Y x
  • R
M y Y j 1 L J Q m Z 9 t I s 9 Q y f W 2 W I g j Z J w 3 O 1 N 8 b K Q m 1 n
  • a
+ n c x S L n t z 8 T + v l 5 i g 7 q V c x
  • l
h k i 4 O B Y n A J s L z C v C Q K a N m F p C q O I 2 K 6 Z j
  • g
g 1 t q i i L c F d / v I q a d e q r l N 1 b y / L j X p e R w F O 4 Q w q 4 M I V N O A G m t A C C g q e 4 R X e B N 6 Q e /
  • Y
z G 6 h v K d E / g D 9 P k D p e S R Q A = = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " G s f Q v I E A V f q R A G p W v W U 6 Q G i N 8 = " > A A A B 9 X i c b V B N T 8 J A F H z F L 8 Q v 1 K O X j c Q E L 6 R V
  • x
x J v H j E x A I R K t k u W 9 i w 3 T a 7 W w 1 p + B 9 e P G i M V / + L N / + N S + l B w U k 2 m c y 8 l z c 7 f s y Z r b 9 b R V W V t f W N 4 q b p a 3 t n d 2 9 8 v 5 B S W J J N Q l E Y 9 k x 8 e K c i a
  • q
5 n m t B N L i k O f 7 Y / v p 7 5 7 U c q F Y v E n Z 7 E 1 A v x U L C A E a y N 9 N A L s R 6 p A N 1 X z y / t 3 6 5 Y t f s D G i Z O D m p Q I 5 m v / z V G Q k C a n Q h G O l u
  • 4
d a y / F U j P C 6 b T U S x S N M R n j I e a K n B I l Z d m q a f
  • x
C g D F E T S P K F R p v 7 e S H G
  • 1
C T z W S W c t G b i f 9 5 3 U Q H d S 9 l I k 4 F W R + K E g 4 h G a V Y A G T F K i + c Q Q T C Q z W R E Z Y Y m J N k W V T A n O 4 p e X S e u s 5 t g 1 5 / a i q j n d R T h C I 6 h C g 5 c Q Q N u
  • A
k u E J D w D K / w Z j 1 Z L 9 a 7 9 T E f L V j 5 z i H 8 g f X 5 A 6 R n k T 8 = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " 1 X g H 5 c j n
  • 4
y c k T X 2
  • b
9 / z e R v 8 I = " > A A A B 9 X i c b V D L S s N A F L 2 p r 1 p f V Z d u B
  • t
Q N y U R i 1 W 3 L i s Y B / Y x j K Z T t q h k m Y m S g l 9 D / c u F D E r f / i z r 9 x k m a h r Q c G D u f c y z 1 z v I g z p W 3 7 2 y q s r W 9 s b h W 3 S z u 7 e / s H 5 c O j j g p j S W i b h D y U P Q 8 r y p m g b c p 7 1 I U h x 4 n H a 9 6 X X q d x + p V C w U d 3
  • W
U T f A Y 8 F 8 R r A 2 s M g w H q i f H R f r d f t 8 2 G 5 Y t f s D G i V O D m p Q I 7 W s P w 1 G I U k D q j Q h G O l +
  • 4
d a T f B U j P C 6 b w i B W N M J n i M e b K n B A l Z t k q e f
  • z
C g j 5 I f S P K F R p v 7 e S H C g 1 C z w z G S W c t l L x f + 8 f q z 9 h p s w E c W a C r I 4 5 M c c 6 R C l F a A R k 5 R
  • P
j M E E 8 l M V k Q m W G K i T V E l U 4 K z / O V V r m
  • O
X b N u b 2 s N B t 5 H U U 4 g V O
  • g
g N X I Q b a E E b C E h 4 h l d 4 s 5 6 s F + v d + l i M F q x 8 5 x j + w P r 8 A a X v k U A = < / l a t e x i t >

Christopher O’Neill (SDSU) Computing delta sets of affine semigroups Feb 10, 2020 22 / 27

slide-154
SLIDE 154

Commutative algebra hiding in the background

S = n1, . . . , nk ⊂ Zd

≥0 (affine)

π : Zk

≥0

− → S A larger example: S = 13, 44, 106, 120. IS = x6

1 x2 4 − x3 3, x2 1 x3 − x3 2, x14 1 x2 − x3 x4, x16 1 − x2 2 x4, x6 1 x4 2 x3 − x3 4

< l a t e x i t s h a 1 _ b a s e 6 4 = " N U W e E n b E i V j O y W k A N B n m W 9 / g v x I = " > A A A B 9 X i c b V D L S s N A F L 2 p r 1 p f U Z d u B
  • t
Q N y W p g l W 3 L i s Y B / Y x j K Z T t q h k m Y m S g l 9 D / c u F D E r f / i z r 9 x m m a h r Q c G D u f c y z 1 z / J g z p R 3 n 2 y q s r W 9 s b h W 3 S z u 7 e / s H 9 u F R W W J J L R F I h 7 J r
  • 8
V 5 U z Q l m a a 2 4 s K Q 5 9 T j v + 5 H r u d x 6 p V C w S d 3
  • a
U y / E I 8 E C R r A 2 k M / x H q s A n R f c S 9 q 5 w O 7 7 F S d D G i V u D k p Q 4 7 m w P 7 q D y O S h F R
  • w
r F S P d e J t Z d i q R n h d F b q J 4 r G m E z w i P Y M F T i k y k u z 1 D N Z p Q h C i J p n t A
  • U
3 9 v p D h U a h r 6 Z j J L u e z N x f + 8 X q K D u p c y E S e a C r I 4 F C Q c 6 Q j N K B D J i n R f G
  • I
J p K Z r I i M s c R E m 6 J K p g R 3 + c u r p F 2 r u k 7 V v b s N + p 5 H U U 4 g V O
  • g
A t X I A b a E I L C E h 4 h l d 4 s 5 6 s F + v d + l i M F q x 8 5 x j + w P r 8 A Z / R k T w = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " c v x g x t x a b S p O H M j q Z h 6 h B M h M r n I = " > A A A B 9 X i c b V D L S g M x F L 3 x W e u r 6 t J N s A h 1 U 2 Z U s M u C G 5 c V 7 A P b s W T S T B u a y Q x J R i l D / 8 O N C X c + i / u / B v T 6 S y 9 U D g c M 6 9 3 J P j x 4 J r 4 z j f a G V 1 b X 1 j s 7 B V 3 N 7 Z 3 d s v H R y 2 d J Q
  • y
p
  • E
p H q + E Q z w S V r G m 4 E 6 8 S K k d A X r O 2 P r 2 d + + 5 E p z S N 5 Z y Y x 8 I y l D z g l B g r P f R C Y k Y 6 w P e V C 7 d 2 1 i + V n a q T A S 8 T N y d l y N H
  • l
7 5 6 g 4 g m I Z O G C q J 1 1 3 V i 4 6 V E G U 4 F m x Z 7 i W Y x
  • W
M y Z F 1 L J Q m Z 9 t I s 9 R S f W m W A g j Z J w 3 O 1 N 8 b K Q m 1 n
  • S
+ n c x S L n
  • z
8 T + v m 5 i g 5 q V c x
  • l
h k s 4 P B Y n A J s K z C v C A K a N m F h C q O I 2 K 6 Y j
  • g
g 1 t q i i L c F d / P I y a Z 1 X X a f q 3 l 6 W 6 7 W 8 j g I c w w l U w I U r q M M N N K A J F B Q 8 w y u 8
  • S
f g t 7 R x 3 x B e U 7 R / A H 6 P M H q P G R Q g = = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " g V H x Y 3 M W F v x P c y e t l F u C P s Q B z k = " > A A A B 9 X i c b V D L S s N A F L 2 p r 1 p f U Z d u B
  • t
Q N y U p
  • l
W 3 L i s Y B / Y x j K Z T t q h k m Y m S g l 9 D / c u F D E r f / i z r 9 x m m a h r Q c G D u f c y z 1 z / J g z p R 3 n 2 y q s r W 9 s b h W 3 S z u 7 e / s H 9 u F R W W J J L R F I h 7 J r
  • 8
V 5 U z Q l m a a 2 4 s K Q 5 9 T j v + 5 H r u d x 6 p V C w S d 3
  • a
U y / E I 8 E C R r A 2 k M / x H q s A n R f q d U u z w d 2 2 a k 6 G d A q c X N S h h z N g f 3 V H Y k C a n Q h G O l e q 4 T a y / F U j P C 6 a z U T x S N M Z n g E e Z K n B I l Z d m q W f
  • z
C h D F E T S P K F R p v 7 e S H G
  • 1
D T z W S W c t m b i / 9 5 v U Q H d S 9 l I k 4 F W R x K E g 4 h G a V 4 C G T F K i + d Q Q T C Q z W R E Z Y 4 m J N k W V T A n u 8 p d X S b t W d Z 2 q e 3 t R b t T z O
  • p
w A q d Q A R e u
  • A
E 3 I Q W E J D w D K / w Z j 1 Z L 9 a 7 9 b E Y L V j 5 z j H 8 g f X 5 A 6 X m k U A = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " q q n s r D D a 3 i H J Y B x U w X + L P Q F k 4 I = " > A A A B 9 X i c b V D L S g M x F L 3 x W e u r 6 t J N s A h 1 U 2 a K Y J c F N y 4 r 2 A e 2 Y 8 m k m T Y k x m S j F K G /
  • c
b F 4 q 4 9 V / c + T e m 1 l
  • 6
4 H A 4 Z x 7 u S f H j w X X x n G + d r 6 x u b W d m G n u L u 3 f 3 B Y O j p u 6 y h R l L V
  • J
C L V 9 Y l m g k v W M t w I 1
  • V
I 6 E v W M e f X M / 9 z i N T m k f y z k x j 5
  • V
k J H n A K T F W e u i H x I x 1 g O 8 r N a d + M S i V n a q T A a 8 S N y d l y N E c l L 7 6 w 4 g m I Z O G C q J 1 z 3 V i 4 6 V E G U 4 F m x X 7 i W Y x
  • R
M y Y j 1 L J Q m Z 9 t I s 9 Q y f W 2 W I g j Z J w 3 O 1 N 8 b K Q m 1 n
  • a
+ n c x S L n t z 8 T + v l 5 i g 7 q V c x
  • l
h k i 4 O B Y n A J s L z C v C Q K a N m F p C q O I 2 K 6 Z j
  • g
g 1 t q i i L c F d / v I q a d e q r l N 1 b y / L j X p e R w F O 4 Q w q 4 M I V N O A G m t A C C g q e 4 R X e B N 6 Q e /
  • Y
z G 6 h v K d E / g D 9 P k D p e S R Q A = = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " G s f Q v I E A V f q R A G p W v W U 6 Q G i N 8 = " > A A A B 9 X i c b V B N T 8 J A F H z F L 8 Q v 1 K O X j c Q E L 6 R V
  • x
x J v H j E x A I R K t k u W 9 i w 3 T a 7 W w 1 p + B 9 e P G i M V / + L N / + N S + l B w U k 2 m c y 8 l z c 7 f s y Z r b 9 b R V W V t f W N 4 q b p a 3 t n d 2 9 8 v 5 B S W J J N Q l E Y 9 k x 8 e K c i a
  • q
5 n m t B N L i k O f 7 Y / v p 7 5 7 U c q F Y v E n Z 7 E 1 A v x U L C A E a y N 9 N A L s R 6 p A N 1 X z y / t 3 6 5 Y t f s D G i Z O D m p Q I 5 m v / z V G Q k C a n Q h G O l u
  • 4
d a y / F U j P C 6 b T U S x S N M R n j I e a K n B I l Z d m q a f
  • x
C g D F E T S P K F R p v 7 e S H G
  • 1
C T z W S W c t G b i f 9 5 3 U Q H d S 9 l I k 4 F W R + K E g 4 h G a V Y A G T F K i + c Q Q T C Q z W R E Z Y Y m J N k W V T A n O 4 p e X S e u s 5 t g 1 5 / a i q j n d R T h C I 6 h C g 5 c Q Q N u
  • A
k u E J D w D K / w Z j 1 Z L 9 a 7 9 T E f L V j 5 z i H 8 g f X 5 A 6 R n k T 8 = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " 1 X g H 5 c j n
  • 4
y c k T X 2
  • b
9 / z e R v 8 I = " > A A A B 9 X i c b V D L S s N A F L 2 p r 1 p f V Z d u B
  • t
Q N y U R i 1 W 3 L i s Y B / Y x j K Z T t q h k m Y m S g l 9 D / c u F D E r f / i z r 9 x k m a h r Q c G D u f c y z 1 z v I g z p W 3 7 2 y q s r W 9 s b h W 3 S z u 7 e / s H 5 c O j j g p j S W i b h D y U P Q 8 r y p m g b c p 7 1 I U h x 4 n H a 9 6 X X q d x + p V C w U d 3
  • W
U T f A Y 8 F 8 R r A 2 s M g w H q i f H R f r d f t 8 2 G 5 Y t f s D G i V O D m p Q I 7 W s P w 1 G I U k D q j Q h G O l +
  • 4
d a T f B U j P C 6 b w i B W N M J n i M e b K n B A l Z t k q e f
  • z
C g j 5 I f S P K F R p v 7 e S H C g 1 C z w z G S W c t l L x f + 8 f q z 9 h p s w E c W a C r I 4 5 M c c 6 R C l F a A R k 5 R
  • P
j M E E 8 l M V k Q m W G K i T V E l U 4 K z / O V V r m
  • O
X b N u b 2 s N B t 5 H U U 4 g V O
  • g
g N X I Q b a E E b C E h 4 h l d 4 s 5 6 s F + v d + l i M F q x 8 5 x j + w P r 8 A a X v k U A = < / l a t e x i t >

Christopher O’Neill (SDSU) Computing delta sets of affine semigroups Feb 10, 2020 22 / 27

slide-155
SLIDE 155

Commutative algebra hiding in the background

S = n1, . . . , nk ⊂ Zd

≥0 (affine)

π : Zk

≥0

− → S A larger example: S = 13, 44, 106, 120. IS = x6

1 x2 4 − x3 3, x2 1 x3 − x3 2, x14 1 x2 − x3 x4, x16 1 − x2 2 x4, x6 1 x4 2 x3 − x3 4

< l a t e x i t s h a 1 _ b a s e 6 4 = " N U W e E n b E i V j O y W k A N B n m W 9 / g v x I = " > A A A B 9 X i c b V D L S s N A F L 2 p r 1 p f U Z d u B
  • t
Q N y W p g l W 3 L i s Y B / Y x j K Z T t q h k m Y m S g l 9 D / c u F D E r f / i z r 9 x m m a h r Q c G D u f c y z 1 z / J g z p R 3 n 2 y q s r W 9 s b h W 3 S z u 7 e / s H 9 u F R W W J J L R F I h 7 J r
  • 8
V 5 U z Q l m a a 2 4 s K Q 5 9 T j v + 5 H r u d x 6 p V C w S d 3
  • a
U y / E I 8 E C R r A 2 k M / x H q s A n R f c S 9 q 5 w O 7 7 F S d D G i V u D k p Q 4 7 m w P 7 q D y O S h F R
  • w
r F S P d e J t Z d i q R n h d F b q J 4 r G m E z w i P Y M F T i k y k u z 1 D N Z p Q h C i J p n t A
  • U
3 9 v p D h U a h r 6 Z j J L u e z N x f + 8 X q K D u p c y E S e a C r I 4 F C Q c 6 Q j N K B D J i n R f G
  • I
J p K Z r I i M s c R E m 6 J K p g R 3 + c u r p F 2 r u k 7 V v b s N + p 5 H U U 4 g V O
  • g
A t X I A b a E I L C E h 4 h l d 4 s 5 6 s F + v d + l i M F q x 8 5 x j + w P r 8 A Z / R k T w = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " c v x g x t x a b S p O H M j q Z h 6 h B M h M r n I = " > A A A B 9 X i c b V D L S g M x F L 3 x W e u r 6 t J N s A h 1 U 2 Z U s M u C G 5 c V 7 A P b s W T S T B u a y Q x J R i l D / 8 O N C X c + i / u / B v T 6 S y 9 U D g c M 6 9 3 J P j x 4 J r 4 z j f a G V 1 b X 1 j s 7 B V 3 N 7 Z 3 d s v H R y 2 d J Q
  • y
p
  • E
p H q + E Q z w S V r G m 4 E 6 8 S K k d A X r O 2 P r 2 d + + 5 E p z S N 5 Z y Y x 8 I y l D z g l B g r P f R C Y k Y 6 w P e V C 7 d 2 1 i + V n a q T A S 8 T N y d l y N H
  • l
7 5 6 g 4 g m I Z O G C q J 1 1 3 V i 4 6 V E G U 4 F m x Z 7 i W Y x
  • W
M y Z F 1 L J Q m Z 9 t I s 9 R S f W m W A g j Z J w 3 O 1 N 8 b K Q m 1 n
  • S
+ n c x S L n
  • z
8 T + v m 5 i g 5 q V c x
  • l
h k s 4 P B Y n A J s K z C v C A K a N m F h C q O I 2 K 6 Y j
  • g
g 1 t q i i L c F d / P I y a Z 1 X X a f q 3 l 6 W 6 7 W 8 j g I c w w l U w I U r q M M N N K A J F B Q 8 w y u 8
  • S
f g t 7 R x 3 x B e U 7 R / A H 6 P M H q P G R Q g = = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " g V H x Y 3 M W F v x P c y e t l F u C P s Q B z k = " > A A A B 9 X i c b V D L S s N A F L 2 p r 1 p f U Z d u B
  • t
Q N y U p
  • l
W 3 L i s Y B / Y x j K Z T t q h k m Y m S g l 9 D / c u F D E r f / i z r 9 x m m a h r Q c G D u f c y z 1 z / J g z p R 3 n 2 y q s r W 9 s b h W 3 S z u 7 e / s H 9 u F R W W J J L R F I h 7 J r
  • 8
V 5 U z Q l m a a 2 4 s K Q 5 9 T j v + 5 H r u d x 6 p V C w S d 3
  • a
U y / E I 8 E C R r A 2 k M / x H q s A n R f q d U u z w d 2 2 a k 6 G d A q c X N S h h z N g f 3 V H Y k C a n Q h G O l e q 4 T a y / F U j P C 6 a z U T x S N M Z n g E e Z K n B I l Z d m q W f
  • z
C h D F E T S P K F R p v 7 e S H G
  • 1
D T z W S W c t m b i / 9 5 v U Q H d S 9 l I k 4 F W R x K E g 4 h G a V 4 C G T F K i + d Q Q T C Q z W R E Z Y 4 m J N k W V T A n u 8 p d X S b t W d Z 2 q e 3 t R b t T z O
  • p
w A q d Q A R e u
  • A
E 3 I Q W E J D w D K / w Z j 1 Z L 9 a 7 9 b E Y L V j 5 z j H 8 g f X 5 A 6 X m k U A = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " q q n s r D D a 3 i H J Y B x U w X + L P Q F k 4 I = " > A A A B 9 X i c b V D L S g M x F L 3 x W e u r 6 t J N s A h 1 U 2 a K Y J c F N y 4 r 2 A e 2 Y 8 m k m T Y k x m S j F K G /
  • c
b F 4 q 4 9 V / c + T e m 1 l
  • 6
4 H A 4 Z x 7 u S f H j w X X x n G + d r 6 x u b W d m G n u L u 3 f 3 B Y O j p u 6 y h R l L V
  • J
C L V 9 Y l m g k v W M t w I 1
  • V
I 6 E v W M e f X M / 9 z i N T m k f y z k x j 5
  • V
k J H n A K T F W e u i H x I x 1 g O 8 r N a d + M S i V n a q T A a 8 S N y d l y N E c l L 7 6 w 4 g m I Z O G C q J 1 z 3 V i 4 6 V E G U 4 F m x X 7 i W Y x
  • R
M y Y j 1 L J Q m Z 9 t I s 9 Q y f W 2 W I g j Z J w 3 O 1 N 8 b K Q m 1 n
  • a
+ n c x S L n t z 8 T + v l 5 i g 7 q V c x
  • l
h k i 4 O B Y n A J s L z C v C Q K a N m F p C q O I 2 K 6 Z j
  • g
g 1 t q i i L c F d / v I q a d e q r l N 1 b y / L j X p e R w F O 4 Q w q 4 M I V N O A G m t A C C g q e 4 R X e B N 6 Q e /
  • Y
z G 6 h v K d E / g D 9 P k D p e S R Q A = = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " G s f Q v I E A V f q R A G p W v W U 6 Q G i N 8 = " > A A A B 9 X i c b V B N T 8 J A F H z F L 8 Q v 1 K O X j c Q E L 6 R V
  • x
x J v H j E x A I R K t k u W 9 i w 3 T a 7 W w 1 p + B 9 e P G i M V / + L N / + N S + l B w U k 2 m c y 8 l z c 7 f s y Z r b 9 b R V W V t f W N 4 q b p a 3 t n d 2 9 8 v 5 B S W J J N Q l E Y 9 k x 8 e K c i a
  • q
5 n m t B N L i k O f 7 Y / v p 7 5 7 U c q F Y v E n Z 7 E 1 A v x U L C A E a y N 9 N A L s R 6 p A N 1 X z y / t 3 6 5 Y t f s D G i Z O D m p Q I 5 m v / z V G Q k C a n Q h G O l u
  • 4
d a y / F U j P C 6 b T U S x S N M R n j I e a K n B I l Z d m q a f
  • x
C g D F E T S P K F R p v 7 e S H G
  • 1
C T z W S W c t G b i f 9 5 3 U Q H d S 9 l I k 4 F W R + K E g 4 h G a V Y A G T F K i + c Q Q T C Q z W R E Z Y Y m J N k W V T A n O 4 p e X S e u s 5 t g 1 5 / a i q j n d R T h C I 6 h C g 5 c Q Q N u
  • A
k u E J D w D K / w Z j 1 Z L 9 a 7 9 T E f L V j 5 z i H 8 g f X 5 A 6 R n k T 8 = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " 1 X g H 5 c j n
  • 4
y c k T X 2
  • b
9 / z e R v 8 I = " > A A A B 9 X i c b V D L S s N A F L 2 p r 1 p f V Z d u B
  • t
Q N y U R i 1 W 3 L i s Y B / Y x j K Z T t q h k m Y m S g l 9 D / c u F D E r f / i z r 9 x k m a h r Q c G D u f c y z 1 z v I g z p W 3 7 2 y q s r W 9 s b h W 3 S z u 7 e / s H 5 c O j j g p j S W i b h D y U P Q 8 r y p m g b c p 7 1 I U h x 4 n H a 9 6 X X q d x + p V C w U d 3
  • W
U T f A Y 8 F 8 R r A 2 s M g w H q i f H R f r d f t 8 2 G 5 Y t f s D G i V O D m p Q I 7 W s P w 1 G I U k D q j Q h G O l +
  • 4
d a T f B U j P C 6 b w i B W N M J n i M e b K n B A l Z t k q e f
  • z
C g j 5 I f S P K F R p v 7 e S H C g 1 C z w z G S W c t l L x f + 8 f q z 9 h p s w E c W a C r I 4 5 M c c 6 R C l F a A R k 5 R
  • P
j M E E 8 l M V k Q m W G K i T V E l U 4 K z / O V V r m
  • O
X b N u b 2 s N B t 5 H U U 4 g V O
  • g
g N X I Q b a E E b C E h 4 h l d 4 s 5 6 s F + v d + l i M F q x 8 5 x j + w P r 8 A a X v k U A = < / l a t e x i t >

Christopher O’Neill (SDSU) Computing delta sets of affine semigroups Feb 10, 2020 22 / 27

slide-156
SLIDE 156

Commutative algebra hiding in the background

S = n1, . . . , nk ⊂ Zd

≥0 (affine)

π : Zk

≥0

− → S A larger example: S = 13, 44, 106, 120. IS = x6

1 x2 4 − x3 3, x2 1 x3 − x3 2, x14 1 x2 − x3 x4, x16 1 − x2 2 x4, x6 1 x4 2 x3 − x3 4

< l a t e x i t s h a 1 _ b a s e 6 4 = " N U W e E n b E i V j O y W k A N B n m W 9 / g v x I = " > A A A B 9 X i c b V D L S s N A F L 2 p r 1 p f U Z d u B
  • t
Q N y W p g l W 3 L i s Y B / Y x j K Z T t q h k m Y m S g l 9 D / c u F D E r f / i z r 9 x m m a h r Q c G D u f c y z 1 z / J g z p R 3 n 2 y q s r W 9 s b h W 3 S z u 7 e / s H 9 u F R W W J J L R F I h 7 J r
  • 8
V 5 U z Q l m a a 2 4 s K Q 5 9 T j v + 5 H r u d x 6 p V C w S d 3
  • a
U y / E I 8 E C R r A 2 k M / x H q s A n R f c S 9 q 5 w O 7 7 F S d D G i V u D k p Q 4 7 m w P 7 q D y O S h F R
  • w
r F S P d e J t Z d i q R n h d F b q J 4 r G m E z w i P Y M F T i k y k u z 1 D N Z p Q h C i J p n t A
  • U
3 9 v p D h U a h r 6 Z j J L u e z N x f + 8 X q K D u p c y E S e a C r I 4 F C Q c 6 Q j N K B D J i n R f G
  • I
J p K Z r I i M s c R E m 6 J K p g R 3 + c u r p F 2 r u k 7 V v b s N + p 5 H U U 4 g V O
  • g
A t X I A b a E I L C E h 4 h l d 4 s 5 6 s F + v d + l i M F q x 8 5 x j + w P r 8 A Z / R k T w = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " c v x g x t x a b S p O H M j q Z h 6 h B M h M r n I = " > A A A B 9 X i c b V D L S g M x F L 3 x W e u r 6 t J N s A h 1 U 2 Z U s M u C G 5 c V 7 A P b s W T S T B u a y Q x J R i l D / 8 O N C X c + i / u / B v T 6 S y 9 U D g c M 6 9 3 J P j x 4 J r 4 z j f a G V 1 b X 1 j s 7 B V 3 N 7 Z 3 d s v H R y 2 d J Q
  • y
p
  • E
p H q + E Q z w S V r G m 4 E 6 8 S K k d A X r O 2 P r 2 d + + 5 E p z S N 5 Z y Y x 8 I y l D z g l B g r P f R C Y k Y 6 w P e V C 7 d 2 1 i + V n a q T A S 8 T N y d l y N H
  • l
7 5 6 g 4 g m I Z O G C q J 1 1 3 V i 4 6 V E G U 4 F m x Z 7 i W Y x
  • W
M y Z F 1 L J Q m Z 9 t I s 9 R S f W m W A g j Z J w 3 O 1 N 8 b K Q m 1 n
  • S
+ n c x S L n
  • z
8 T + v m 5 i g 5 q V c x
  • l
h k s 4 P B Y n A J s K z C v C A K a N m F h C q O I 2 K 6 Y j
  • g
g 1 t q i i L c F d / P I y a Z 1 X X a f q 3 l 6 W 6 7 W 8 j g I c w w l U w I U r q M M N N K A J F B Q 8 w y u 8
  • S
f g t 7 R x 3 x B e U 7 R / A H 6 P M H q P G R Q g = = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " g V H x Y 3 M W F v x P c y e t l F u C P s Q B z k = " > A A A B 9 X i c b V D L S s N A F L 2 p r 1 p f U Z d u B
  • t
Q N y U p
  • l
W 3 L i s Y B / Y x j K Z T t q h k m Y m S g l 9 D / c u F D E r f / i z r 9 x m m a h r Q c G D u f c y z 1 z / J g z p R 3 n 2 y q s r W 9 s b h W 3 S z u 7 e / s H 9 u F R W W J J L R F I h 7 J r
  • 8
V 5 U z Q l m a a 2 4 s K Q 5 9 T j v + 5 H r u d x 6 p V C w S d 3
  • a
U y / E I 8 E C R r A 2 k M / x H q s A n R f q d U u z w d 2 2 a k 6 G d A q c X N S h h z N g f 3 V H Y k C a n Q h G O l e q 4 T a y / F U j P C 6 a z U T x S N M Z n g E e Z K n B I l Z d m q W f
  • z
C h D F E T S P K F R p v 7 e S H G
  • 1
D T z W S W c t m b i / 9 5 v U Q H d S 9 l I k 4 F W R x K E g 4 h G a V 4 C G T F K i + d Q Q T C Q z W R E Z Y 4 m J N k W V T A n u 8 p d X S b t W d Z 2 q e 3 t R b t T z O
  • p
w A q d Q A R e u
  • A
E 3 I Q W E J D w D K / w Z j 1 Z L 9 a 7 9 b E Y L V j 5 z j H 8 g f X 5 A 6 X m k U A = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " q q n s r D D a 3 i H J Y B x U w X + L P Q F k 4 I = " > A A A B 9 X i c b V D L S g M x F L 3 x W e u r 6 t J N s A h 1 U 2 a K Y J c F N y 4 r 2 A e 2 Y 8 m k m T Y k x m S j F K G /
  • c
b F 4 q 4 9 V / c + T e m 1 l
  • 6
4 H A 4 Z x 7 u S f H j w X X x n G + d r 6 x u b W d m G n u L u 3 f 3 B Y O j p u 6 y h R l L V
  • J
C L V 9 Y l m g k v W M t w I 1
  • V
I 6 E v W M e f X M / 9 z i N T m k f y z k x j 5
  • V
k J H n A K T F W e u i H x I x 1 g O 8 r N a d + M S i V n a q T A a 8 S N y d l y N E c l L 7 6 w 4 g m I Z O G C q J 1 z 3 V i 4 6 V E G U 4 F m x X 7 i W Y x
  • R
M y Y j 1 L J Q m Z 9 t I s 9 Q y f W 2 W I g j Z J w 3 O 1 N 8 b K Q m 1 n
  • a
+ n c x S L n t z 8 T + v l 5 i g 7 q V c x
  • l
h k i 4 O B Y n A J s L z C v C Q K a N m F p C q O I 2 K 6 Z j
  • g
g 1 t q i i L c F d / v I q a d e q r l N 1 b y / L j X p e R w F O 4 Q w q 4 M I V N O A G m t A C C g q e 4 R X e B N 6 Q e /
  • Y
z G 6 h v K d E / g D 9 P k D p e S R Q A = = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " G s f Q v I E A V f q R A G p W v W U 6 Q G i N 8 = " > A A A B 9 X i c b V B N T 8 J A F H z F L 8 Q v 1 K O X j c Q E L 6 R V
  • x
x J v H j E x A I R K t k u W 9 i w 3 T a 7 W w 1 p + B 9 e P G i M V / + L N / + N S + l B w U k 2 m c y 8 l z c 7 f s y Z r b 9 b R V W V t f W N 4 q b p a 3 t n d 2 9 8 v 5 B S W J J N Q l E Y 9 k x 8 e K c i a
  • q
5 n m t B N L i k O f 7 Y / v p 7 5 7 U c q F Y v E n Z 7 E 1 A v x U L C A E a y N 9 N A L s R 6 p A N 1 X z y / t 3 6 5 Y t f s D G i Z O D m p Q I 5 m v / z V G Q k C a n Q h G O l u
  • 4
d a y / F U j P C 6 b T U S x S N M R n j I e a K n B I l Z d m q a f
  • x
C g D F E T S P K F R p v 7 e S H G
  • 1
C T z W S W c t G b i f 9 5 3 U Q H d S 9 l I k 4 F W R + K E g 4 h G a V Y A G T F K i + c Q Q T C Q z W R E Z Y Y m J N k W V T A n O 4 p e X S e u s 5 t g 1 5 / a i q j n d R T h C I 6 h C g 5 c Q Q N u
  • A
k u E J D w D K / w Z j 1 Z L 9 a 7 9 T E f L V j 5 z i H 8 g f X 5 A 6 R n k T 8 = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " 1 X g H 5 c j n
  • 4
y c k T X 2
  • b
9 / z e R v 8 I = " > A A A B 9 X i c b V D L S s N A F L 2 p r 1 p f V Z d u B
  • t
Q N y U R i 1 W 3 L i s Y B / Y x j K Z T t q h k m Y m S g l 9 D / c u F D E r f / i z r 9 x k m a h r Q c G D u f c y z 1 z v I g z p W 3 7 2 y q s r W 9 s b h W 3 S z u 7 e / s H 5 c O j j g p j S W i b h D y U P Q 8 r y p m g b c p 7 1 I U h x 4 n H a 9 6 X X q d x + p V C w U d 3
  • W
U T f A Y 8 F 8 R r A 2 s M g w H q i f H R f r d f t 8 2 G 5 Y t f s D G i V O D m p Q I 7 W s P w 1 G I U k D q j Q h G O l +
  • 4
d a T f B U j P C 6 b w i B W N M J n i M e b K n B A l Z t k q e f
  • z
C g j 5 I f S P K F R p v 7 e S H C g 1 C z w z G S W c t l L x f + 8 f q z 9 h p s w E c W a C r I 4 5 M c c 6 R C l F a A R k 5 R
  • P
j M E E 8 l M V k Q m W G K i T V E l U 4 K z / O V V r m
  • O
X b N u b 2 s N B t 5 H U U 4 g V O
  • g
g N X I Q b a E E b C E h 4 h l d 4 s 5 6 s F + v d + l i M F q x 8 5 x j + w P r 8 A a X v k U A = < / l a t e x i t >

Christopher O’Neill (SDSU) Computing delta sets of affine semigroups Feb 10, 2020 22 / 27

slide-157
SLIDE 157

Commutative algebra hiding in the background

S = n1, . . . , nk ⊂ Zd

≥0 (affine)

π : Zk

≥0

− → S A larger example: S = 13, 44, 106, 120. IS = x6

1 x2 4 − x3 3, x2 1 x3 − x3 2, x14 1 x2 − x3 x4, x16 1 − x2 2 x4, x6 1 x4 2 x3 − x3 4

< l a t e x i t s h a 1 _ b a s e 6 4 = " N U W e E n b E i V j O y W k A N B n m W 9 / g v x I = " > A A A B 9 X i c b V D L S s N A F L 2 p r 1 p f U Z d u B
  • t
Q N y W p g l W 3 L i s Y B / Y x j K Z T t q h k m Y m S g l 9 D / c u F D E r f / i z r 9 x m m a h r Q c G D u f c y z 1 z / J g z p R 3 n 2 y q s r W 9 s b h W 3 S z u 7 e / s H 9 u F R W W J J L R F I h 7 J r
  • 8
V 5 U z Q l m a a 2 4 s K Q 5 9 T j v + 5 H r u d x 6 p V C w S d 3
  • a
U y / E I 8 E C R r A 2 k M / x H q s A n R f c S 9 q 5 w O 7 7 F S d D G i V u D k p Q 4 7 m w P 7 q D y O S h F R
  • w
r F S P d e J t Z d i q R n h d F b q J 4 r G m E z w i P Y M F T i k y k u z 1 D N Z p Q h C i J p n t A
  • U
3 9 v p D h U a h r 6 Z j J L u e z N x f + 8 X q K D u p c y E S e a C r I 4 F C Q c 6 Q j N K B D J i n R f G
  • I
J p K Z r I i M s c R E m 6 J K p g R 3 + c u r p F 2 r u k 7 V v b s N + p 5 H U U 4 g V O
  • g
A t X I A b a E I L C E h 4 h l d 4 s 5 6 s F + v d + l i M F q x 8 5 x j + w P r 8 A Z / R k T w = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " c v x g x t x a b S p O H M j q Z h 6 h B M h M r n I = " > A A A B 9 X i c b V D L S g M x F L 3 x W e u r 6 t J N s A h 1 U 2 Z U s M u C G 5 c V 7 A P b s W T S T B u a y Q x J R i l D / 8 O N C X c + i / u / B v T 6 S y 9 U D g c M 6 9 3 J P j x 4 J r 4 z j f a G V 1 b X 1 j s 7 B V 3 N 7 Z 3 d s v H R y 2 d J Q
  • y
p
  • E
p H q + E Q z w S V r G m 4 E 6 8 S K k d A X r O 2 P r 2 d + + 5 E p z S N 5 Z y Y x 8 I y l D z g l B g r P f R C Y k Y 6 w P e V C 7 d 2 1 i + V n a q T A S 8 T N y d l y N H
  • l
7 5 6 g 4 g m I Z O G C q J 1 1 3 V i 4 6 V E G U 4 F m x Z 7 i W Y x
  • W
M y Z F 1 L J Q m Z 9 t I s 9 R S f W m W A g j Z J w 3 O 1 N 8 b K Q m 1 n
  • S
+ n c x S L n
  • z
8 T + v m 5 i g 5 q V c x
  • l
h k s 4 P B Y n A J s K z C v C A K a N m F h C q O I 2 K 6 Y j
  • g
g 1 t q i i L c F d / P I y a Z 1 X X a f q 3 l 6 W 6 7 W 8 j g I c w w l U w I U r q M M N N K A J F B Q 8 w y u 8
  • S
f g t 7 R x 3 x B e U 7 R / A H 6 P M H q P G R Q g = = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " g V H x Y 3 M W F v x P c y e t l F u C P s Q B z k = " > A A A B 9 X i c b V D L S s N A F L 2 p r 1 p f U Z d u B
  • t
Q N y U p
  • l
W 3 L i s Y B / Y x j K Z T t q h k m Y m S g l 9 D / c u F D E r f / i z r 9 x m m a h r Q c G D u f c y z 1 z / J g z p R 3 n 2 y q s r W 9 s b h W 3 S z u 7 e / s H 9 u F R W W J J L R F I h 7 J r
  • 8
V 5 U z Q l m a a 2 4 s K Q 5 9 T j v + 5 H r u d x 6 p V C w S d 3
  • a
U y / E I 8 E C R r A 2 k M / x H q s A n R f q d U u z w d 2 2 a k 6 G d A q c X N S h h z N g f 3 V H Y k C a n Q h G O l e q 4 T a y / F U j P C 6 a z U T x S N M Z n g E e Z K n B I l Z d m q W f
  • z
C h D F E T S P K F R p v 7 e S H G
  • 1
D T z W S W c t m b i / 9 5 v U Q H d S 9 l I k 4 F W R x K E g 4 h G a V 4 C G T F K i + d Q Q T C Q z W R E Z Y 4 m J N k W V T A n u 8 p d X S b t W d Z 2 q e 3 t R b t T z O
  • p
w A q d Q A R e u
  • A
E 3 I Q W E J D w D K / w Z j 1 Z L 9 a 7 9 b E Y L V j 5 z j H 8 g f X 5 A 6 X m k U A = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " q q n s r D D a 3 i H J Y B x U w X + L P Q F k 4 I = " > A A A B 9 X i c b V D L S g M x F L 3 x W e u r 6 t J N s A h 1 U 2 a K Y J c F N y 4 r 2 A e 2 Y 8 m k m T Y k x m S j F K G /
  • c
b F 4 q 4 9 V / c + T e m 1 l
  • 6
4 H A 4 Z x 7 u S f H j w X X x n G + d r 6 x u b W d m G n u L u 3 f 3 B Y O j p u 6 y h R l L V
  • J
C L V 9 Y l m g k v W M t w I 1
  • V
I 6 E v W M e f X M / 9 z i N T m k f y z k x j 5
  • V
k J H n A K T F W e u i H x I x 1 g O 8 r N a d + M S i V n a q T A a 8 S N y d l y N E c l L 7 6 w 4 g m I Z O G C q J 1 z 3 V i 4 6 V E G U 4 F m x X 7 i W Y x
  • R
M y Y j 1 L J Q m Z 9 t I s 9 Q y f W 2 W I g j Z J w 3 O 1 N 8 b K Q m 1 n
  • a
+ n c x S L n t z 8 T + v l 5 i g 7 q V c x
  • l
h k i 4 O B Y n A J s L z C v C Q K a N m F p C q O I 2 K 6 Z j
  • g
g 1 t q i i L c F d / v I q a d e q r l N 1 b y / L j X p e R w F O 4 Q w q 4 M I V N O A G m t A C C g q e 4 R X e B N 6 Q e /
  • Y
z G 6 h v K d E / g D 9 P k D p e S R Q A = = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " G s f Q v I E A V f q R A G p W v W U 6 Q G i N 8 = " > A A A B 9 X i c b V B N T 8 J A F H z F L 8 Q v 1 K O X j c Q E L 6 R V
  • x
x J v H j E x A I R K t k u W 9 i w 3 T a 7 W w 1 p + B 9 e P G i M V / + L N / + N S + l B w U k 2 m c y 8 l z c 7 f s y Z r b 9 b R V W V t f W N 4 q b p a 3 t n d 2 9 8 v 5 B S W J J N Q l E Y 9 k x 8 e K c i a
  • q
5 n m t B N L i k O f 7 Y / v p 7 5 7 U c q F Y v E n Z 7 E 1 A v x U L C A E a y N 9 N A L s R 6 p A N 1 X z y / t 3 6 5 Y t f s D G i Z O D m p Q I 5 m v / z V G Q k C a n Q h G O l u
  • 4
d a y / F U j P C 6 b T U S x S N M R n j I e a K n B I l Z d m q a f
  • x
C g D F E T S P K F R p v 7 e S H G
  • 1
C T z W S W c t G b i f 9 5 3 U Q H d S 9 l I k 4 F W R + K E g 4 h G a V Y A G T F K i + c Q Q T C Q z W R E Z Y Y m J N k W V T A n O 4 p e X S e u s 5 t g 1 5 / a i q j n d R T h C I 6 h C g 5 c Q Q N u
  • A
k u E J D w D K / w Z j 1 Z L 9 a 7 9 T E f L V j 5 z i H 8 g f X 5 A 6 R n k T 8 = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " 1 X g H 5 c j n
  • 4
y c k T X 2
  • b
9 / z e R v 8 I = " > A A A B 9 X i c b V D L S s N A F L 2 p r 1 p f V Z d u B
  • t
Q N y U R i 1 W 3 L i s Y B / Y x j K Z T t q h k m Y m S g l 9 D / c u F D E r f / i z r 9 x k m a h r Q c G D u f c y z 1 z v I g z p W 3 7 2 y q s r W 9 s b h W 3 S z u 7 e / s H 5 c O j j g p j S W i b h D y U P Q 8 r y p m g b c p 7 1 I U h x 4 n H a 9 6 X X q d x + p V C w U d 3
  • W
U T f A Y 8 F 8 R r A 2 s M g w H q i f H R f r d f t 8 2 G 5 Y t f s D G i V O D m p Q I 7 W s P w 1 G I U k D q j Q h G O l +
  • 4
d a T f B U j P C 6 b w i B W N M J n i M e b K n B A l Z t k q e f
  • z
C g j 5 I f S P K F R p v 7 e S H C g 1 C z w z G S W c t l L x f + 8 f q z 9 h p s w E c W a C r I 4 5 M c c 6 R C l F a A R k 5 R
  • P
j M E E 8 l M V k Q m W G K i T V E l U 4 K z / O V V r m
  • O
X b N u b 2 s N B t 5 H U U 4 g V O
  • g
g N X I Q b a E E b C E h 4 h l d 4 s 5 6 s F + v d + l i M F q x 8 5 x j + w P r 8 A a X v k U A = < / l a t e x i t >

Christopher O’Neill (SDSU) Computing delta sets of affine semigroups Feb 10, 2020 22 / 27

slide-158
SLIDE 158

Commutative algebra hiding in the background

S = n1, . . . , nk ⊂ Zd

≥0 (affine)

π : Zk

≥0

− → S A larger example: S = 13, 44, 106, 120. IS = x6

1 x2 4 − x3 3, x2 1 x3 − x3 2, x14 1 x2 − x3 x4, x16 1 − x2 2 x4, x6 1 x4 2 x3 − x3 4

< l a t e x i t s h a 1 _ b a s e 6 4 = " N U W e E n b E i V j O y W k A N B n m W 9 / g v x I = " > A A A B 9 X i c b V D L S s N A F L 2 p r 1 p f U Z d u B
  • t
Q N y W p g l W 3 L i s Y B / Y x j K Z T t q h k m Y m S g l 9 D / c u F D E r f / i z r 9 x m m a h r Q c G D u f c y z 1 z / J g z p R 3 n 2 y q s r W 9 s b h W 3 S z u 7 e / s H 9 u F R W W J J L R F I h 7 J r
  • 8
V 5 U z Q l m a a 2 4 s K Q 5 9 T j v + 5 H r u d x 6 p V C w S d 3
  • a
U y / E I 8 E C R r A 2 k M / x H q s A n R f c S 9 q 5 w O 7 7 F S d D G i V u D k p Q 4 7 m w P 7 q D y O S h F R
  • w
r F S P d e J t Z d i q R n h d F b q J 4 r G m E z w i P Y M F T i k y k u z 1 D N Z p Q h C i J p n t A
  • U
3 9 v p D h U a h r 6 Z j J L u e z N x f + 8 X q K D u p c y E S e a C r I 4 F C Q c 6 Q j N K B D J i n R f G
  • I
J p K Z r I i M s c R E m 6 J K p g R 3 + c u r p F 2 r u k 7 V v b s N + p 5 H U U 4 g V O
  • g
A t X I A b a E I L C E h 4 h l d 4 s 5 6 s F + v d + l i M F q x 8 5 x j + w P r 8 A Z / R k T w = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " c v x g x t x a b S p O H M j q Z h 6 h B M h M r n I = " > A A A B 9 X i c b V D L S g M x F L 3 x W e u r 6 t J N s A h 1 U 2 Z U s M u C G 5 c V 7 A P b s W T S T B u a y Q x J R i l D / 8 O N C X c + i / u / B v T 6 S y 9 U D g c M 6 9 3 J P j x 4 J r 4 z j f a G V 1 b X 1 j s 7 B V 3 N 7 Z 3 d s v H R y 2 d J Q
  • y
p
  • E
p H q + E Q z w S V r G m 4 E 6 8 S K k d A X r O 2 P r 2 d + + 5 E p z S N 5 Z y Y x 8 I y l D z g l B g r P f R C Y k Y 6 w P e V C 7 d 2 1 i + V n a q T A S 8 T N y d l y N H
  • l
7 5 6 g 4 g m I Z O G C q J 1 1 3 V i 4 6 V E G U 4 F m x Z 7 i W Y x
  • W
M y Z F 1 L J Q m Z 9 t I s 9 R S f W m W A g j Z J w 3 O 1 N 8 b K Q m 1 n
  • S
+ n c x S L n
  • z
8 T + v m 5 i g 5 q V c x
  • l
h k s 4 P B Y n A J s K z C v C A K a N m F h C q O I 2 K 6 Y j
  • g
g 1 t q i i L c F d / P I y a Z 1 X X a f q 3 l 6 W 6 7 W 8 j g I c w w l U w I U r q M M N N K A J F B Q 8 w y u 8
  • S
f g t 7 R x 3 x B e U 7 R / A H 6 P M H q P G R Q g = = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " g V H x Y 3 M W F v x P c y e t l F u C P s Q B z k = " > A A A B 9 X i c b V D L S s N A F L 2 p r 1 p f U Z d u B
  • t
Q N y U p
  • l
W 3 L i s Y B / Y x j K Z T t q h k m Y m S g l 9 D / c u F D E r f / i z r 9 x m m a h r Q c G D u f c y z 1 z / J g z p R 3 n 2 y q s r W 9 s b h W 3 S z u 7 e / s H 9 u F R W W J J L R F I h 7 J r
  • 8
V 5 U z Q l m a a 2 4 s K Q 5 9 T j v + 5 H r u d x 6 p V C w S d 3
  • a
U y / E I 8 E C R r A 2 k M / x H q s A n R f q d U u z w d 2 2 a k 6 G d A q c X N S h h z N g f 3 V H Y k C a n Q h G O l e q 4 T a y / F U j P C 6 a z U T x S N M Z n g E e Z K n B I l Z d m q W f
  • z
C h D F E T S P K F R p v 7 e S H G
  • 1
D T z W S W c t m b i / 9 5 v U Q H d S 9 l I k 4 F W R x K E g 4 h G a V 4 C G T F K i + d Q Q T C Q z W R E Z Y 4 m J N k W V T A n u 8 p d X S b t W d Z 2 q e 3 t R b t T z O
  • p
w A q d Q A R e u
  • A
E 3 I Q W E J D w D K / w Z j 1 Z L 9 a 7 9 b E Y L V j 5 z j H 8 g f X 5 A 6 X m k U A = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " q q n s r D D a 3 i H J Y B x U w X + L P Q F k 4 I = " > A A A B 9 X i c b V D L S g M x F L 3 x W e u r 6 t J N s A h 1 U 2 a K Y J c F N y 4 r 2 A e 2 Y 8 m k m T Y k x m S j F K G /
  • c
b F 4 q 4 9 V / c + T e m 1 l
  • 6
4 H A 4 Z x 7 u S f H j w X X x n G + d r 6 x u b W d m G n u L u 3 f 3 B Y O j p u 6 y h R l L V
  • J
C L V 9 Y l m g k v W M t w I 1
  • V
I 6 E v W M e f X M / 9 z i N T m k f y z k x j 5
  • V
k J H n A K T F W e u i H x I x 1 g O 8 r N a d + M S i V n a q T A a 8 S N y d l y N E c l L 7 6 w 4 g m I Z O G C q J 1 z 3 V i 4 6 V E G U 4 F m x X 7 i W Y x
  • R
M y Y j 1 L J Q m Z 9 t I s 9 Q y f W 2 W I g j Z J w 3 O 1 N 8 b K Q m 1 n
  • a
+ n c x S L n t z 8 T + v l 5 i g 7 q V c x
  • l
h k i 4 O B Y n A J s L z C v C Q K a N m F p C q O I 2 K 6 Z j
  • g
g 1 t q i i L c F d / v I q a d e q r l N 1 b y / L j X p e R w F O 4 Q w q 4 M I V N O A G m t A C C g q e 4 R X e B N 6 Q e /
  • Y
z G 6 h v K d E / g D 9 P k D p e S R Q A = = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " G s f Q v I E A V f q R A G p W v W U 6 Q G i N 8 = " > A A A B 9 X i c b V B N T 8 J A F H z F L 8 Q v 1 K O X j c Q E L 6 R V
  • x
x J v H j E x A I R K t k u W 9 i w 3 T a 7 W w 1 p + B 9 e P G i M V / + L N / + N S + l B w U k 2 m c y 8 l z c 7 f s y Z r b 9 b R V W V t f W N 4 q b p a 3 t n d 2 9 8 v 5 B S W J J N Q l E Y 9 k x 8 e K c i a
  • q
5 n m t B N L i k O f 7 Y / v p 7 5 7 U c q F Y v E n Z 7 E 1 A v x U L C A E a y N 9 N A L s R 6 p A N 1 X z y / t 3 6 5 Y t f s D G i Z O D m p Q I 5 m v / z V G Q k C a n Q h G O l u
  • 4
d a y / F U j P C 6 b T U S x S N M R n j I e a K n B I l Z d m q a f
  • x
C g D F E T S P K F R p v 7 e S H G
  • 1
C T z W S W c t G b i f 9 5 3 U Q H d S 9 l I k 4 F W R + K E g 4 h G a V Y A G T F K i + c Q Q T C Q z W R E Z Y Y m J N k W V T A n O 4 p e X S e u s 5 t g 1 5 / a i q j n d R T h C I 6 h C g 5 c Q Q N u
  • A
k u E J D w D K / w Z j 1 Z L 9 a 7 9 T E f L V j 5 z i H 8 g f X 5 A 6 R n k T 8 = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " Z q a l 2 M q q H k u
  • m
D / 3 x O Z M l P u C 4 Q w = " > A A A B 9 X i c b V D L S g M x F L 1 T X 7 W + q i 7 d B I t Q N y U j g l W 3 L i s Y B / Y j i W T Z t r Q T G Z I M k
  • Z
+ h 9 u X C j i 1 n 9 x 5 9 + Y T m e h r Q c C h 3 P u 5 Z 4 c P x Z c G 4 y / n c L a + s b m V n G 7 t L O 7 t 3 9 Q P j x q 6 y h R l L V
  • J
C L V 9 Y l m g k v W M t w I 1
  • V
I 6 E v W M e f X M / 9 z i N T m k f y z k x j 5
  • V
k J H n A K T F W e u i H x I x 1 g O 6 r d Y z P B + U K r u E M a J W 4 O a l A j u a g / N U f R j Q J m T R U E K 1 7 L
  • 6
N l x J l O B V s V u
  • n
m s W E T s i I 9 S y V J G T a S 7 P U M 3 R m l S E K I m W f N C h T f 2 + k J N R 6 G v p 2 M k u 5 7 M 3 F / 7 x e Y
  • K
6 l 3 I Z J 4 Z J u j g U J A K Z C M r Q E O u G D V i a g m h i t u s i I 6 J I t T Y
  • k
q 2 B H f 5 y 6 u k f V F z c c 2 9 v a w 6 n k d R T i B U 6 i C C 1 f Q g B t
  • Q
g s
  • K
H i G V 3 h z n p w X 5 9 3 5 W I w W n H z n G P 7 A + f w B
  • u
a R P g = = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " 1 X g H 5 c j n
  • 4
y c k T X 2
  • b
9 / z e R v 8 I = " > A A A B 9 X i c b V D L S s N A F L 2 p r 1 p f V Z d u B
  • t
Q N y U R i 1 W 3 L i s Y B / Y x j K Z T t q h k m Y m S g l 9 D / c u F D E r f / i z r 9 x k m a h r Q c G D u f c y z 1 z v I g z p W 3 7 2 y q s r W 9 s b h W 3 S z u 7 e / s H 5 c O j j g p j S W i b h D y U P Q 8 r y p m g b c p 7 1 I U h x 4 n H a 9 6 X X q d x + p V C w U d 3
  • W
U T f A Y 8 F 8 R r A 2 s M g w H q i f H R f r d f t 8 2 G 5 Y t f s D G i V O D m p Q I 7 W s P w 1 G I U k D q j Q h G O l +
  • 4
d a T f B U j P C 6 b w i B W N M J n i M e b K n B A l Z t k q e f
  • z
C g j 5 I f S P K F R p v 7 e S H C g 1 C z w z G S W c t l L x f + 8 f q z 9 h p s w E c W a C r I 4 5 M c c 6 R C l F a A R k 5 R
  • P
j M E E 8 l M V k Q m W G K i T V E l U 4 K z / O V V r m
  • O
X b N u b 2 s N B t 5 H U U 4 g V O
  • g
g N X I Q b a E E b C E h 4 h l d 4 s 5 6 s F + v d + l i M F q x 8 5 x j + w P r 8 A a X v k U A = < / l a t e x i t >

Christopher O’Neill (SDSU) Computing delta sets of affine semigroups Feb 10, 2020 22 / 27

slide-159
SLIDE 159

Commutative algebra hiding in the background

S = n1, . . . , nk ⊂ Zd

≥0 (affine)

π : Zk

≥0

− → S A larger example: S = 13, 44, 106, 120. IS = x6

1 x2 4 − x3 3, x2 1 x3 − x3 2, x14 1 x2 − x3 x4, x16 1 − x2 2 x4, x6 1 x4 2 x3 − x3 4

< l a t e x i t s h a 1 _ b a s e 6 4 = " N U W e E n b E i V j O y W k A N B n m W 9 / g v x I = " > A A A B 9 X i c b V D L S s N A F L 2 p r 1 p f U Z d u B
  • t
Q N y W p g l W 3 L i s Y B / Y x j K Z T t q h k m Y m S g l 9 D / c u F D E r f / i z r 9 x m m a h r Q c G D u f c y z 1 z / J g z p R 3 n 2 y q s r W 9 s b h W 3 S z u 7 e / s H 9 u F R W W J J L R F I h 7 J r
  • 8
V 5 U z Q l m a a 2 4 s K Q 5 9 T j v + 5 H r u d x 6 p V C w S d 3
  • a
U y / E I 8 E C R r A 2 k M / x H q s A n R f c S 9 q 5 w O 7 7 F S d D G i V u D k p Q 4 7 m w P 7 q D y O S h F R
  • w
r F S P d e J t Z d i q R n h d F b q J 4 r G m E z w i P Y M F T i k y k u z 1 D N Z p Q h C i J p n t A
  • U
3 9 v p D h U a h r 6 Z j J L u e z N x f + 8 X q K D u p c y E S e a C r I 4 F C Q c 6 Q j N K B D J i n R f G
  • I
J p K Z r I i M s c R E m 6 J K p g R 3 + c u r p F 2 r u k 7 V v b s N + p 5 H U U 4 g V O
  • g
A t X I A b a E I L C E h 4 h l d 4 s 5 6 s F + v d + l i M F q x 8 5 x j + w P r 8 A Z / R k T w = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " c v x g x t x a b S p O H M j q Z h 6 h B M h M r n I = " > A A A B 9 X i c b V D L S g M x F L 3 x W e u r 6 t J N s A h 1 U 2 Z U s M u C G 5 c V 7 A P b s W T S T B u a y Q x J R i l D / 8 O N C X c + i / u / B v T 6 S y 9 U D g c M 6 9 3 J P j x 4 J r 4 z j f a G V 1 b X 1 j s 7 B V 3 N 7 Z 3 d s v H R y 2 d J Q
  • y
p
  • E
p H q + E Q z w S V r G m 4 E 6 8 S K k d A X r O 2 P r 2 d + + 5 E p z S N 5 Z y Y x 8 I y l D z g l B g r P f R C Y k Y 6 w P e V C 7 d 2 1 i + V n a q T A S 8 T N y d l y N H
  • l
7 5 6 g 4 g m I Z O G C q J 1 1 3 V i 4 6 V E G U 4 F m x Z 7 i W Y x
  • W
M y Z F 1 L J Q m Z 9 t I s 9 R S f W m W A g j Z J w 3 O 1 N 8 b K Q m 1 n
  • S
+ n c x S L n
  • z
8 T + v m 5 i g 5 q V c x
  • l
h k s 4 P B Y n A J s K z C v C A K a N m F h C q O I 2 K 6 Y j
  • g
g 1 t q i i L c F d / P I y a Z 1 X X a f q 3 l 6 W 6 7 W 8 j g I c w w l U w I U r q M M N N K A J F B Q 8 w y u 8
  • S
f g t 7 R x 3 x B e U 7 R / A H 6 P M H q P G R Q g = = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " g V H x Y 3 M W F v x P c y e t l F u C P s Q B z k = " > A A A B 9 X i c b V D L S s N A F L 2 p r 1 p f U Z d u B
  • t
Q N y U p
  • l
W 3 L i s Y B / Y x j K Z T t q h k m Y m S g l 9 D / c u F D E r f / i z r 9 x m m a h r Q c G D u f c y z 1 z / J g z p R 3 n 2 y q s r W 9 s b h W 3 S z u 7 e / s H 9 u F R W W J J L R F I h 7 J r
  • 8
V 5 U z Q l m a a 2 4 s K Q 5 9 T j v + 5 H r u d x 6 p V C w S d 3
  • a
U y / E I 8 E C R r A 2 k M / x H q s A n R f q d U u z w d 2 2 a k 6 G d A q c X N S h h z N g f 3 V H Y k C a n Q h G O l e q 4 T a y / F U j P C 6 a z U T x S N M Z n g E e Z K n B I l Z d m q W f
  • z
C h D F E T S P K F R p v 7 e S H G
  • 1
D T z W S W c t m b i / 9 5 v U Q H d S 9 l I k 4 F W R x K E g 4 h G a V 4 C G T F K i + d Q Q T C Q z W R E Z Y 4 m J N k W V T A n u 8 p d X S b t W d Z 2 q e 3 t R b t T z O
  • p
w A q d Q A R e u
  • A
E 3 I Q W E J D w D K / w Z j 1 Z L 9 a 7 9 b E Y L V j 5 z j H 8 g f X 5 A 6 X m k U A = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " q q n s r D D a 3 i H J Y B x U w X + L P Q F k 4 I = " > A A A B 9 X i c b V D L S g M x F L 3 x W e u r 6 t J N s A h 1 U 2 a K Y J c F N y 4 r 2 A e 2 Y 8 m k m T Y k x m S j F K G /
  • c
b F 4 q 4 9 V / c + T e m 1 l
  • 6
4 H A 4 Z x 7 u S f H j w X X x n G + d r 6 x u b W d m G n u L u 3 f 3 B Y O j p u 6 y h R l L V
  • J
C L V 9 Y l m g k v W M t w I 1
  • V
I 6 E v W M e f X M / 9 z i N T m k f y z k x j 5
  • V
k J H n A K T F W e u i H x I x 1 g O 8 r N a d + M S i V n a q T A a 8 S N y d l y N E c l L 7 6 w 4 g m I Z O G C q J 1 z 3 V i 4 6 V E G U 4 F m x X 7 i W Y x
  • R
M y Y j 1 L J Q m Z 9 t I s 9 Q y f W 2 W I g j Z J w 3 O 1 N 8 b K Q m 1 n
  • a
+ n c x S L n t z 8 T + v l 5 i g 7 q V c x
  • l
h k i 4 O B Y n A J s L z C v C Q K a N m F p C q O I 2 K 6 Z j
  • g
g 1 t q i i L c F d / v I q a d e q r l N 1 b y / L j X p e R w F O 4 Q w q 4 M I V N O A G m t A C C g q e 4 R X e B N 6 Q e /
  • Y
z G 6 h v K d E / g D 9 P k D p e S R Q A = = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " G s f Q v I E A V f q R A G p W v W U 6 Q G i N 8 = " > A A A B 9 X i c b V B N T 8 J A F H z F L 8 Q v 1 K O X j c Q E L 6 R V
  • x
x J v H j E x A I R K t k u W 9 i w 3 T a 7 W w 1 p + B 9 e P G i M V / + L N / + N S + l B w U k 2 m c y 8 l z c 7 f s y Z r b 9 b R V W V t f W N 4 q b p a 3 t n d 2 9 8 v 5 B S W J J N Q l E Y 9 k x 8 e K c i a
  • q
5 n m t B N L i k O f 7 Y / v p 7 5 7 U c q F Y v E n Z 7 E 1 A v x U L C A E a y N 9 N A L s R 6 p A N 1 X z y / t 3 6 5 Y t f s D G i Z O D m p Q I 5 m v / z V G Q k C a n Q h G O l u
  • 4
d a y / F U j P C 6 b T U S x S N M R n j I e a K n B I l Z d m q a f
  • x
C g D F E T S P K F R p v 7 e S H G
  • 1
C T z W S W c t G b i f 9 5 3 U Q H d S 9 l I k 4 F W R + K E g 4 h G a V Y A G T F K i + c Q Q T C Q z W R E Z Y Y m J N k W V T A n O 4 p e X S e u s 5 t g 1 5 / a i q j n d R T h C I 6 h C g 5 c Q Q N u
  • A
k u E J D w D K / w Z j 1 Z L 9 a 7 9 T E f L V j 5 z i H 8 g f X 5 A 6 R n k T 8 = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " Z q a l 2 M q q H k u
  • m
D / 3 x O Z M l P u C 4 Q w = " > A A A B 9 X i c b V D L S g M x F L 1 T X 7 W + q i 7 d B I t Q N y U j g l W 3 L i s Y B / Y j i W T Z t r Q T G Z I M k
  • Z
+ h 9 u X C j i 1 n 9 x 5 9 + Y T m e h r Q c C h 3 P u 5 Z 4 c P x Z c G 4 y / n c L a + s b m V n G 7 t L O 7 t 3 9 Q P j x q 6 y h R l L V
  • J
C L V 9 Y l m g k v W M t w I 1
  • V
I 6 E v W M e f X M / 9 z i N T m k f y z k x j 5
  • V
k J H n A K T F W e u i H x I x 1 g O 6 r d Y z P B + U K r u E M a J W 4 O a l A j u a g / N U f R j Q J m T R U E K 1 7 L
  • 6
N l x J l O B V s V u
  • n
m s W E T s i I 9 S y V J G T a S 7 P U M 3 R m l S E K I m W f N C h T f 2 + k J N R 6 G v p 2 M k u 5 7 M 3 F / 7 x e Y
  • K
6 l 3 I Z J 4 Z J u j g U J A K Z C M r Q E O u G D V i a g m h i t u s i I 6 J I t T Y
  • k
q 2 B H f 5 y 6 u k f V F z c c 2 9 v a w 6 n k d R T i B U 6 i C C 1 f Q g B t
  • Q
g s
  • K
H i G V 3 h z n p w X 5 9 3 5 W I w W n H z n G P 7 A + f w B
  • u
a R P g = = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " 1 X g H 5 c j n
  • 4
y c k T X 2
  • b
9 / z e R v 8 I = " > A A A B 9 X i c b V D L S s N A F L 2 p r 1 p f V Z d u B
  • t
Q N y U R i 1 W 3 L i s Y B / Y x j K Z T t q h k m Y m S g l 9 D / c u F D E r f / i z r 9 x k m a h r Q c G D u f c y z 1 z v I g z p W 3 7 2 y q s r W 9 s b h W 3 S z u 7 e / s H 5 c O j j g p j S W i b h D y U P Q 8 r y p m g b c p 7 1 I U h x 4 n H a 9 6 X X q d x + p V C w U d 3
  • W
U T f A Y 8 F 8 R r A 2 s M g w H q i f H R f r d f t 8 2 G 5 Y t f s D G i V O D m p Q I 7 W s P w 1 G I U k D q j Q h G O l +
  • 4
d a T f B U j P C 6 b w i B W N M J n i M e b K n B A l Z t k q e f
  • z
C g j 5 I f S P K F R p v 7 e S H C g 1 C z w z G S W c t l L x f + 8 f q z 9 h p s w E c W a C r I 4 5 M c c 6 R C l F a A R k 5 R
  • P
j M E E 8 l M V k Q m W G K i T V E l U 4 K z / O V V r m
  • O
X b N u b 2 s N B t 5 H U U 4 g V O
  • g
g N X I Q b a E E b C E h 4 h l d 4 s 5 6 s F + v d + l i M F q x 8 5 x j + w P r 8 A a X v k U A = < / l a t e x i t >

Christopher O’Neill (SDSU) Computing delta sets of affine semigroups Feb 10, 2020 22 / 27

slide-160
SLIDE 160

Commutative algebra hiding in the background

S = n1, . . . , nk ⊂ Zd

≥0 (affine)

π : Zk

≥0

− → S A larger example: S = 13, 44, 106, 120. IS = x6

1 x2 4 − x3 3, x2 1 x3 − x3 2, x14 1 x2 − x3 x4, x16 1 − x2 2 x4, x6 1 x4 2 x3 − x3 4

< l a t e x i t s h a 1 _ b a s e 6 4 = " N U W e E n b E i V j O y W k A N B n m W 9 / g v x I = " > A A A B 9 X i c b V D L S s N A F L 2 p r 1 p f U Z d u B
  • t
Q N y W p g l W 3 L i s Y B / Y x j K Z T t q h k m Y m S g l 9 D / c u F D E r f / i z r 9 x m m a h r Q c G D u f c y z 1 z / J g z p R 3 n 2 y q s r W 9 s b h W 3 S z u 7 e / s H 9 u F R W W J J L R F I h 7 J r
  • 8
V 5 U z Q l m a a 2 4 s K Q 5 9 T j v + 5 H r u d x 6 p V C w S d 3
  • a
U y / E I 8 E C R r A 2 k M / x H q s A n R f c S 9 q 5 w O 7 7 F S d D G i V u D k p Q 4 7 m w P 7 q D y O S h F R
  • w
r F S P d e J t Z d i q R n h d F b q J 4 r G m E z w i P Y M F T i k y k u z 1 D N Z p Q h C i J p n t A
  • U
3 9 v p D h U a h r 6 Z j J L u e z N x f + 8 X q K D u p c y E S e a C r I 4 F C Q c 6 Q j N K B D J i n R f G
  • I
J p K Z r I i M s c R E m 6 J K p g R 3 + c u r p F 2 r u k 7 V v b s N + p 5 H U U 4 g V O
  • g
A t X I A b a E I L C E h 4 h l d 4 s 5 6 s F + v d + l i M F q x 8 5 x j + w P r 8 A Z / R k T w = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " c v x g x t x a b S p O H M j q Z h 6 h B M h M r n I = " > A A A B 9 X i c b V D L S g M x F L 3 x W e u r 6 t J N s A h 1 U 2 Z U s M u C G 5 c V 7 A P b s W T S T B u a y Q x J R i l D / 8 O N C X c + i / u / B v T 6 S y 9 U D g c M 6 9 3 J P j x 4 J r 4 z j f a G V 1 b X 1 j s 7 B V 3 N 7 Z 3 d s v H R y 2 d J Q
  • y
p
  • E
p H q + E Q z w S V r G m 4 E 6 8 S K k d A X r O 2 P r 2 d + + 5 E p z S N 5 Z y Y x 8 I y l D z g l B g r P f R C Y k Y 6 w P e V C 7 d 2 1 i + V n a q T A S 8 T N y d l y N H
  • l
7 5 6 g 4 g m I Z O G C q J 1 1 3 V i 4 6 V E G U 4 F m x Z 7 i W Y x
  • W
M y Z F 1 L J Q m Z 9 t I s 9 R S f W m W A g j Z J w 3 O 1 N 8 b K Q m 1 n
  • S
+ n c x S L n
  • z
8 T + v m 5 i g 5 q V c x
  • l
h k s 4 P B Y n A J s K z C v C A K a N m F h C q O I 2 K 6 Y j
  • g
g 1 t q i i L c F d / P I y a Z 1 X X a f q 3 l 6 W 6 7 W 8 j g I c w w l U w I U r q M M N N K A J F B Q 8 w y u 8
  • S
f g t 7 R x 3 x B e U 7 R / A H 6 P M H q P G R Q g = = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " g V H x Y 3 M W F v x P c y e t l F u C P s Q B z k = " > A A A B 9 X i c b V D L S s N A F L 2 p r 1 p f U Z d u B
  • t
Q N y U p
  • l
W 3 L i s Y B / Y x j K Z T t q h k m Y m S g l 9 D / c u F D E r f / i z r 9 x m m a h r Q c G D u f c y z 1 z / J g z p R 3 n 2 y q s r W 9 s b h W 3 S z u 7 e / s H 9 u F R W W J J L R F I h 7 J r
  • 8
V 5 U z Q l m a a 2 4 s K Q 5 9 T j v + 5 H r u d x 6 p V C w S d 3
  • a
U y / E I 8 E C R r A 2 k M / x H q s A n R f q d U u z w d 2 2 a k 6 G d A q c X N S h h z N g f 3 V H Y k C a n Q h G O l e q 4 T a y / F U j P C 6 a z U T x S N M Z n g E e Z K n B I l Z d m q W f
  • z
C h D F E T S P K F R p v 7 e S H G
  • 1
D T z W S W c t m b i / 9 5 v U Q H d S 9 l I k 4 F W R x K E g 4 h G a V 4 C G T F K i + d Q Q T C Q z W R E Z Y 4 m J N k W V T A n u 8 p d X S b t W d Z 2 q e 3 t R b t T z O
  • p
w A q d Q A R e u
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E 3 I Q W E J D w D K / w Z j 1 Z L 9 a 7 9 b E Y L V j 5 z j H 8 g f X 5 A 6 X m k U A = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " q q n s r D D a 3 i H J Y B x U w X + L P Q F k 4 I = " > A A A B 9 X i c b V D L S g M x F L 3 x W e u r 6 t J N s A h 1 U 2 a K Y J c F N y 4 r 2 A e 2 Y 8 m k m T Y k x m S j F K G /
  • c
b F 4 q 4 9 V / c + T e m 1 l
  • 6
4 H A 4 Z x 7 u S f H j w X X x n G + d r 6 x u b W d m G n u L u 3 f 3 B Y O j p u 6 y h R l L V
  • J
C L V 9 Y l m g k v W M t w I 1
  • V
I 6 E v W M e f X M / 9 z i N T m k f y z k x j 5
  • V
k J H n A K T F W e u i H x I x 1 g O 8 r N a d + M S i V n a q T A a 8 S N y d l y N E c l L 7 6 w 4 g m I Z O G C q J 1 z 3 V i 4 6 V E G U 4 F m x X 7 i W Y x
  • R
M y Y j 1 L J Q m Z 9 t I s 9 Q y f W 2 W I g j Z J w 3 O 1 N 8 b K Q m 1 n
  • a
+ n c x S L n t z 8 T + v l 5 i g 7 q V c x
  • l
h k i 4 O B Y n A J s L z C v C Q K a N m F p C q O I 2 K 6 Z j
  • g
g 1 t q i i L c F d / v I q a d e q r l N 1 b y / L j X p e R w F O 4 Q w q 4 M I V N O A G m t A C C g q e 4 R X e B N 6 Q e /
  • Y
z G 6 h v K d E / g D 9 P k D p e S R Q A = = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " G s f Q v I E A V f q R A G p W v W U 6 Q G i N 8 = " > A A A B 9 X i c b V B N T 8 J A F H z F L 8 Q v 1 K O X j c Q E L 6 R V
  • x
x J v H j E x A I R K t k u W 9 i w 3 T a 7 W w 1 p + B 9 e P G i M V / + L N / + N S + l B w U k 2 m c y 8 l z c 7 f s y Z r b 9 b R V W V t f W N 4 q b p a 3 t n d 2 9 8 v 5 B S W J J N Q l E Y 9 k x 8 e K c i a
  • q
5 n m t B N L i k O f 7 Y / v p 7 5 7 U c q F Y v E n Z 7 E 1 A v x U L C A E a y N 9 N A L s R 6 p A N 1 X z y / t 3 6 5 Y t f s D G i Z O D m p Q I 5 m v / z V G Q k C a n Q h G O l u
  • 4
d a y / F U j P C 6 b T U S x S N M R n j I e a K n B I l Z d m q a f
  • x
C g D F E T S P K F R p v 7 e S H G
  • 1
C T z W S W c t G b i f 9 5 3 U Q H d S 9 l I k 4 F W R + K E g 4 h G a V Y A G T F K i + c Q Q T C Q z W R E Z Y Y m J N k W V T A n O 4 p e X S e u s 5 t g 1 5 / a i q j n d R T h C I 6 h C g 5 c Q Q N u
  • A
k u E J D w D K / w Z j 1 Z L 9 a 7 9 T E f L V j 5 z i H 8 g f X 5 A 6 R n k T 8 = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " Z q a l 2 M q q H k u
  • m
D / 3 x O Z M l P u C 4 Q w = " > A A A B 9 X i c b V D L S g M x F L 1 T X 7 W + q i 7 d B I t Q N y U j g l W 3 L i s Y B / Y j i W T Z t r Q T G Z I M k
  • Z
+ h 9 u X C j i 1 n 9 x 5 9 + Y T m e h r Q c C h 3 P u 5 Z 4 c P x Z c G 4 y / n c L a + s b m V n G 7 t L O 7 t 3 9 Q P j x q 6 y h R l L V
  • J
C L V 9 Y l m g k v W M t w I 1
  • V
I 6 E v W M e f X M / 9 z i N T m k f y z k x j 5
  • V
k J H n A K T F W e u i H x I x 1 g O 6 r d Y z P B + U K r u E M a J W 4 O a l A j u a g / N U f R j Q J m T R U E K 1 7 L
  • 6
N l x J l O B V s V u
  • n
m s W E T s i I 9 S y V J G T a S 7 P U M 3 R m l S E K I m W f N C h T f 2 + k J N R 6 G v p 2 M k u 5 7 M 3 F / 7 x e Y
  • K
6 l 3 I Z J 4 Z J u j g U J A K Z C M r Q E O u G D V i a g m h i t u s i I 6 J I t T Y
  • k
q 2 B H f 5 y 6 u k f V F z c c 2 9 v a w 6 n k d R T i B U 6 i C C 1 f Q g B t
  • Q
g s
  • K
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  • u
a R P g = = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " 1 X g H 5 c j n
  • 4
y c k T X 2
  • b
9 / z e R v 8 I = " > A A A B 9 X i c b V D L S s N A F L 2 p r 1 p f V Z d u B
  • t
Q N y U R i 1 W 3 L i s Y B / Y x j K Z T t q h k m Y m S g l 9 D / c u F D E r f / i z r 9 x k m a h r Q c G D u f c y z 1 z v I g z p W 3 7 2 y q s r W 9 s b h W 3 S z u 7 e / s H 5 c O j j g p j S W i b h D y U P Q 8 r y p m g b c p 7 1 I U h x 4 n H a 9 6 X X q d x + p V C w U d 3
  • W
U T f A Y 8 F 8 R r A 2 s M g w H q i f H R f r d f t 8 2 G 5 Y t f s D G i V O D m p Q I 7 W s P w 1 G I U k D q j Q h G O l +
  • 4
d a T f B U j P C 6 b w i B W N M J n i M e b K n B A l Z t k q e f
  • z
C g j 5 I f S P K F R p v 7 e S H C g 1 C z w z G S W c t l L x f + 8 f q z 9 h p s w E c W a C r I 4 5 M c c 6 R C l F a A R k 5 R
  • P
j M E E 8 l M V k Q m W G K i T V E l U 4 K z / O V V r m
  • O
X b N u b 2 s N B t 5 H U U 4 g V O
  • g
g N X I Q b a E E b C E h 4 h l d 4 s 5 6 s F + v d + l i M F q x 8 5 x j + w P r 8 A a X v k U A = < / l a t e x i t >

Christopher O’Neill (SDSU) Computing delta sets of affine semigroups Feb 10, 2020 22 / 27

slide-161
SLIDE 161

Commutative algebra hiding in the background

S = n1, . . . , nk ⊂ Zd

≥0 (affine)

π : Zk

≥0

− → S A larger example: S = 13, 44, 106, 120. IS = x6

1 x2 4 − x3 3, x2 1 x3 − x3 2, x14 1 x2 − x3 x4, x16 1 − x2 2 x4, x6 1 x4 2 x3 − x3 4

< l a t e x i t s h a 1 _ b a s e 6 4 = " N U W e E n b E i V j O y W k A N B n m W 9 / g v x I = " > A A A B 9 X i c b V D L S s N A F L 2 p r 1 p f U Z d u B
  • t
Q N y W p g l W 3 L i s Y B / Y x j K Z T t q h k m Y m S g l 9 D / c u F D E r f / i z r 9 x m m a h r Q c G D u f c y z 1 z / J g z p R 3 n 2 y q s r W 9 s b h W 3 S z u 7 e / s H 9 u F R W W J J L R F I h 7 J r
  • 8
V 5 U z Q l m a a 2 4 s K Q 5 9 T j v + 5 H r u d x 6 p V C w S d 3
  • a
U y / E I 8 E C R r A 2 k M / x H q s A n R f c S 9 q 5 w O 7 7 F S d D G i V u D k p Q 4 7 m w P 7 q D y O S h F R
  • w
r F S P d e J t Z d i q R n h d F b q J 4 r G m E z w i P Y M F T i k y k u z 1 D N Z p Q h C i J p n t A
  • U
3 9 v p D h U a h r 6 Z j J L u e z N x f + 8 X q K D u p c y E S e a C r I 4 F C Q c 6 Q j N K B D J i n R f G
  • I
J p K Z r I i M s c R E m 6 J K p g R 3 + c u r p F 2 r u k 7 V v b s N + p 5 H U U 4 g V O
  • g
A t X I A b a E I L C E h 4 h l d 4 s 5 6 s F + v d + l i M F q x 8 5 x j + w P r 8 A Z / R k T w = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " c v x g x t x a b S p O H M j q Z h 6 h B M h M r n I = " > A A A B 9 X i c b V D L S g M x F L 3 x W e u r 6 t J N s A h 1 U 2 Z U s M u C G 5 c V 7 A P b s W T S T B u a y Q x J R i l D / 8 O N C X c + i / u / B v T 6 S y 9 U D g c M 6 9 3 J P j x 4 J r 4 z j f a G V 1 b X 1 j s 7 B V 3 N 7 Z 3 d s v H R y 2 d J Q
  • y
p
  • E
p H q + E Q z w S V r G m 4 E 6 8 S K k d A X r O 2 P r 2 d + + 5 E p z S N 5 Z y Y x 8 I y l D z g l B g r P f R C Y k Y 6 w P e V C 7 d 2 1 i + V n a q T A S 8 T N y d l y N H
  • l
7 5 6 g 4 g m I Z O G C q J 1 1 3 V i 4 6 V E G U 4 F m x Z 7 i W Y x
  • W
M y Z F 1 L J Q m Z 9 t I s 9 R S f W m W A g j Z J w 3 O 1 N 8 b K Q m 1 n
  • S
+ n c x S L n
  • z
8 T + v m 5 i g 5 q V c x
  • l
h k s 4 P B Y n A J s K z C v C A K a N m F h C q O I 2 K 6 Y j
  • g
g 1 t q i i L c F d / P I y a Z 1 X X a f q 3 l 6 W 6 7 W 8 j g I c w w l U w I U r q M M N N K A J F B Q 8 w y u 8
  • S
f g t 7 R x 3 x B e U 7 R / A H 6 P M H q P G R Q g = = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " g V H x Y 3 M W F v x P c y e t l F u C P s Q B z k = " > A A A B 9 X i c b V D L S s N A F L 2 p r 1 p f U Z d u B
  • t
Q N y U p
  • l
W 3 L i s Y B / Y x j K Z T t q h k m Y m S g l 9 D / c u F D E r f / i z r 9 x m m a h r Q c G D u f c y z 1 z / J g z p R 3 n 2 y q s r W 9 s b h W 3 S z u 7 e / s H 9 u F R W W J J L R F I h 7 J r
  • 8
V 5 U z Q l m a a 2 4 s K Q 5 9 T j v + 5 H r u d x 6 p V C w S d 3
  • a
U y / E I 8 E C R r A 2 k M / x H q s A n R f q d U u z w d 2 2 a k 6 G d A q c X N S h h z N g f 3 V H Y k C a n Q h G O l e q 4 T a y / F U j P C 6 a z U T x S N M Z n g E e Z K n B I l Z d m q W f
  • z
C h D F E T S P K F R p v 7 e S H G
  • 1
D T z W S W c t m b i / 9 5 v U Q H d S 9 l I k 4 F W R x K E g 4 h G a V 4 C G T F K i + d Q Q T C Q z W R E Z Y 4 m J N k W V T A n u 8 p d X S b t W d Z 2 q e 3 t R b t T z O
  • p
w A q d Q A R e u
  • A
E 3 I Q W E J D w D K / w Z j 1 Z L 9 a 7 9 b E Y L V j 5 z j H 8 g f X 5 A 6 X m k U A = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " q q n s r D D a 3 i H J Y B x U w X + L P Q F k 4 I = " > A A A B 9 X i c b V D L S g M x F L 3 x W e u r 6 t J N s A h 1 U 2 a K Y J c F N y 4 r 2 A e 2 Y 8 m k m T Y k x m S j F K G /
  • c
b F 4 q 4 9 V / c + T e m 1 l
  • 6
4 H A 4 Z x 7 u S f H j w X X x n G + d r 6 x u b W d m G n u L u 3 f 3 B Y O j p u 6 y h R l L V
  • J
C L V 9 Y l m g k v W M t w I 1
  • V
I 6 E v W M e f X M / 9 z i N T m k f y z k x j 5
  • V
k J H n A K T F W e u i H x I x 1 g O 8 r N a d + M S i V n a q T A a 8 S N y d l y N E c l L 7 6 w 4 g m I Z O G C q J 1 z 3 V i 4 6 V E G U 4 F m x X 7 i W Y x
  • R
M y Y j 1 L J Q m Z 9 t I s 9 Q y f W 2 W I g j Z J w 3 O 1 N 8 b K Q m 1 n
  • a
+ n c x S L n t z 8 T + v l 5 i g 7 q V c x
  • l
h k i 4 O B Y n A J s L z C v C Q K a N m F p C q O I 2 K 6 Z j
  • g
g 1 t q i i L c F d / v I q a d e q r l N 1 b y / L j X p e R w F O 4 Q w q 4 M I V N O A G m t A C C g q e 4 R X e B N 6 Q e /
  • Y
z G 6 h v K d E / g D 9 P k D p e S R Q A = = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " G s f Q v I E A V f q R A G p W v W U 6 Q G i N 8 = " > A A A B 9 X i c b V B N T 8 J A F H z F L 8 Q v 1 K O X j c Q E L 6 R V
  • x
x J v H j E x A I R K t k u W 9 i w 3 T a 7 W w 1 p + B 9 e P G i M V / + L N / + N S + l B w U k 2 m c y 8 l z c 7 f s y Z r b 9 b R V W V t f W N 4 q b p a 3 t n d 2 9 8 v 5 B S W J J N Q l E Y 9 k x 8 e K c i a
  • q
5 n m t B N L i k O f 7 Y / v p 7 5 7 U c q F Y v E n Z 7 E 1 A v x U L C A E a y N 9 N A L s R 6 p A N 1 X z y / t 3 6 5 Y t f s D G i Z O D m p Q I 5 m v / z V G Q k C a n Q h G O l u
  • 4
d a y / F U j P C 6 b T U S x S N M R n j I e a K n B I l Z d m q a f
  • x
C g D F E T S P K F R p v 7 e S H G
  • 1
C T z W S W c t G b i f 9 5 3 U Q H d S 9 l I k 4 F W R + K E g 4 h G a V Y A G T F K i + c Q Q T C Q z W R E Z Y Y m J N k W V T A n O 4 p e X S e u s 5 t g 1 5 / a i q j n d R T h C I 6 h C g 5 c Q Q N u
  • A
k u E J D w D K / w Z j 1 Z L 9 a 7 9 T E f L V j 5 z i H 8 g f X 5 A 6 R n k T 8 = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " Z q a l 2 M q q H k u
  • m
D / 3 x O Z M l P u C 4 Q w = " > A A A B 9 X i c b V D L S g M x F L 1 T X 7 W + q i 7 d B I t Q N y U j g l W 3 L i s Y B / Y j i W T Z t r Q T G Z I M k
  • Z
+ h 9 u X C j i 1 n 9 x 5 9 + Y T m e h r Q c C h 3 P u 5 Z 4 c P x Z c G 4 y / n c L a + s b m V n G 7 t L O 7 t 3 9 Q P j x q 6 y h R l L V
  • J
C L V 9 Y l m g k v W M t w I 1
  • V
I 6 E v W M e f X M / 9 z i N T m k f y z k x j 5
  • V
k J H n A K T F W e u i H x I x 1 g O 6 r d Y z P B + U K r u E M a J W 4 O a l A j u a g / N U f R j Q J m T R U E K 1 7 L
  • 6
N l x J l O B V s V u
  • n
m s W E T s i I 9 S y V J G T a S 7 P U M 3 R m l S E K I m W f N C h T f 2 + k J N R 6 G v p 2 M k u 5 7 M 3 F / 7 x e Y
  • K
6 l 3 I Z J 4 Z J u j g U J A K Z C M r Q E O u G D V i a g m h i t u s i I 6 J I t T Y
  • k
q 2 B H f 5 y 6 u k f V F z c c 2 9 v a w 6 n k d R T i B U 6 i C C 1 f Q g B t
  • Q
g s
  • K
H i G V 3 h z n p w X 5 9 3 5 W I w W n H z n G P 7 A + f w B
  • u
a R P g = = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " 1 X g H 5 c j n
  • 4
y c k T X 2
  • b
9 / z e R v 8 I = " > A A A B 9 X i c b V D L S s N A F L 2 p r 1 p f V Z d u B
  • t
Q N y U R i 1 W 3 L i s Y B / Y x j K Z T t q h k m Y m S g l 9 D / c u F D E r f / i z r 9 x k m a h r Q c G D u f c y z 1 z v I g z p W 3 7 2 y q s r W 9 s b h W 3 S z u 7 e / s H 5 c O j j g p j S W i b h D y U P Q 8 r y p m g b c p 7 1 I U h x 4 n H a 9 6 X X q d x + p V C w U d 3
  • W
U T f A Y 8 F 8 R r A 2 s M g w H q i f H R f r d f t 8 2 G 5 Y t f s D G i V O D m p Q I 7 W s P w 1 G I U k D q j Q h G O l +
  • 4
d a T f B U j P C 6 b w i B W N M J n i M e b K n B A l Z t k q e f
  • z
C g j 5 I f S P K F R p v 7 e S H C g 1 C z w z G S W c t l L x f + 8 f q z 9 h p s w E c W a C r I 4 5 M c c 6 R C l F a A R k 5 R
  • P
j M E E 8 l M V k Q m W G K i T V E l U 4 K z / O V V r m
  • O
X b N u b 2 s N B t 5 H U U 4 g V O
  • g
g N X I Q b a E E b C E h 4 h l d 4 s 5 6 s F + v d + l i M F q x 8 5 x j + w P r 8 A a X v k U A = < / l a t e x i t >

Christopher O’Neill (SDSU) Computing delta sets of affine semigroups Feb 10, 2020 22 / 27

slide-162
SLIDE 162

The delta set via commutative algebra

π : Zk

≥0

− → S = n1, . . . , nk a − → a1n1 + · · · + aknk ϕ : k[x1, . . . , xk] − → k[w] xi − → wni IS = ker(ϕ) = xa − xb : π(a) = π(b)

Christopher O’Neill (SDSU) Computing delta sets of affine semigroups Feb 10, 2020 23 / 27

slide-163
SLIDE 163

The delta set via commutative algebra

π : Zk

≥0

− → S = n1, . . . , nk a − → a1n1 + · · · + aknk ϕ : k[x1, . . . , xk] − → k[w] xi − → wni IS = ker(ϕ) = xa − xb : π(a) = π(b) Ij = xa − xb : π(a) = π(b) and

  • |a| − |b|
  • ≤ j ⊂ IS

Christopher O’Neill (SDSU) Computing delta sets of affine semigroups Feb 10, 2020 23 / 27

slide-164
SLIDE 164

The delta set via commutative algebra

π : Zk

≥0

− → S = n1, . . . , nk a − → a1n1 + · · · + aknk ϕ : k[x1, . . . , xk] − → k[w] xi − → wni IS = ker(ϕ) = xa − xb : π(a) = π(b) Ij = xa − xb : π(a) = π(b) and

  • |a| − |b|
  • ≤ j ⊂ IS

Idea: only connect some of the factorizations

Christopher O’Neill (SDSU) Computing delta sets of affine semigroups Feb 10, 2020 23 / 27

slide-165
SLIDE 165

The delta set via commutative algebra

π : Zk

≥0

− → S = n1, . . . , nk a − → a1n1 + · · · + aknk ϕ : k[x1, . . . , xk] − → k[w] xi − → wni IS = ker(ϕ) = xa − xb : π(a) = π(b) Ij = xa − xb : π(a) = π(b) and

  • |a| − |b|
  • ≤ j ⊂ IS

Idea: only connect some of the factorizations

Christopher O’Neill (SDSU) Computing delta sets of affine semigroups Feb 10, 2020 23 / 27

slide-166
SLIDE 166

The delta set via commutative algebra

π : Zk

≥0

− → S = n1, . . . , nk a − → a1n1 + · · · + aknk ϕ : k[x1, . . . , xk] − → k[w] xi − → wni IS = ker(ϕ) = xa − xb : π(a) = π(b) Ij = xa − xb : π(a) = π(b) and

  • |a| − |b|
  • ≤ j ⊂ IS

Idea: only connect some of the factorizations I0 ⊂ I1 ⊂ I2 ⊂ I3 ⊂ I4 ⊂ · · · ⊂ IS

Christopher O’Neill (SDSU) Computing delta sets of affine semigroups Feb 10, 2020 23 / 27

slide-167
SLIDE 167

The delta set via commutative algebra

π : Zk

≥0

− → S = n1, . . . , nk a − → a1n1 + · · · + aknk ϕ : k[x1, . . . , xk] − → k[w] xi − → wni IS = ker(ϕ) = xa − xb : π(a) = π(b) Ij = xa − xb : π(a) = π(b) and

  • |a| − |b|
  • ≤ j ⊂ IS

Idea: only connect some of the factorizations I0 ⊂ I1 ⊂ I2 ⊂ I3 ⊂ I4 ⊂ · · · ⊂ IS

Christopher O’Neill (SDSU) Computing delta sets of affine semigroups Feb 10, 2020 23 / 27

slide-168
SLIDE 168

The delta set via commutative algebra

π : Zk

≥0

− → S = n1, . . . , nk a − → a1n1 + · · · + aknk ϕ : k[x1, . . . , xk] − → k[w] xi − → wni IS = ker(ϕ) = xa − xb : π(a) = π(b) Ij = xa − xb : π(a) = π(b) and

  • |a| − |b|
  • ≤ j ⊂ IS

Idea: only connect some of the factorizations I0 ⊂ I1 ⊂ I2 ⊂ I3 ⊂ I4 ⊂ · · · ⊂ IS Example: S6, 9, 20 IS = x3 − y2, x4y4 − z3 ⊂ k[x, y, z]

Christopher O’Neill (SDSU) Computing delta sets of affine semigroups Feb 10, 2020 23 / 27

slide-169
SLIDE 169

The delta set via commutative algebra

π : Zk

≥0

− → S = n1, . . . , nk a − → a1n1 + · · · + aknk ϕ : k[x1, . . . , xk] − → k[w] xi − → wni IS = ker(ϕ) = xa − xb : π(a) = π(b) Ij = xa − xb : π(a) = π(b) and

  • |a| − |b|
  • ≤ j ⊂ IS

Idea: only connect some of the factorizations I0 ⊂ I1 ⊂ I2 ⊂ I3 ⊂ I4 ⊂ · · · ⊂ IS Example: S6, 9, 20 IS = x3 − y2, x4y4 − z3 ⊂ k[x, y, z] Z(244): connected components: 28

Christopher O’Neill (SDSU) Computing delta sets of affine semigroups Feb 10, 2020 23 / 27

slide-170
SLIDE 170

The delta set via commutative algebra

π : Zk

≥0

− → S = n1, . . . , nk a − → a1n1 + · · · + aknk ϕ : k[x1, . . . , xk] − → k[w] xi − → wni IS = ker(ϕ) = xa − xb : π(a) = π(b) Ij = xa − xb : π(a) = π(b) and

  • |a| − |b|
  • ≤ j ⊂ IS

Idea: only connect some of the factorizations I0 ⊂ I1 ⊂ I2 ⊂ I3 ⊂ I4 ⊂ · · · ⊂ IS Example: S6, 9, 20 IS = x3 − y2, x4y4 − z3 ⊂ k[x, y, z] Z(244): connected components: 21

Christopher O’Neill (SDSU) Computing delta sets of affine semigroups Feb 10, 2020 23 / 27

slide-171
SLIDE 171

The delta set via commutative algebra

π : Zk

≥0

− → S = n1, . . . , nk a − → a1n1 + · · · + aknk ϕ : k[x1, . . . , xk] − → k[w] xi − → wni IS = ker(ϕ) = xa − xb : π(a) = π(b) Ij = xa − xb : π(a) = π(b) and

  • |a| − |b|
  • ≤ j ⊂ IS

Idea: only connect some of the factorizations I0 ⊂ I1 ⊂ I2 ⊂ I3 ⊂ I4 ⊂ · · · ⊂ IS Example: S6, 9, 20 IS = x3 − y2, x4y4 − z3 ⊂ k[x, y, z] Z(244): connected components: 21

Christopher O’Neill (SDSU) Computing delta sets of affine semigroups Feb 10, 2020 23 / 27

slide-172
SLIDE 172

The delta set via commutative algebra

π : Zk

≥0

− → S = n1, . . . , nk a − → a1n1 + · · · + aknk ϕ : k[x1, . . . , xk] − → k[w] xi − → wni IS = ker(ϕ) = xa − xb : π(a) = π(b) Ij = xa − xb : π(a) = π(b) and

  • |a| − |b|
  • ≤ j ⊂ IS

Idea: only connect some of the factorizations I0 ⊂ I1 ⊂ I2 ⊂ I3 ⊂ I4 ⊂ · · · ⊂ IS Example: S6, 9, 20 IS = x3 − y2, x4y4 − z3 ⊂ k[x, y, z] Z(244): connected components: 2

Christopher O’Neill (SDSU) Computing delta sets of affine semigroups Feb 10, 2020 23 / 27

slide-173
SLIDE 173

The delta set via commutative algebra

π : Zk

≥0

− → S = n1, . . . , nk a − → a1n1 + · · · + aknk ϕ : k[x1, . . . , xk] − → k[w] xi − → wni IS = ker(ϕ) = xa − xb : π(a) = π(b) Ij = xa − xb : π(a) = π(b) and

  • |a| − |b|
  • ≤ j ⊂ IS

Idea: only connect some of the factorizations I0 ⊂ I1 ⊂ I2 ⊂ I3 ⊂ I4 ⊂ · · · ⊂ IS Example: S6, 9, 20 IS = x3 − y2, x4y4 − z3 ⊂ k[x, y, z] Z(244): connected components: 2

Christopher O’Neill (SDSU) Computing delta sets of affine semigroups Feb 10, 2020 23 / 27

slide-174
SLIDE 174

The delta set via commutative algebra

π : Zk

≥0

− → S = n1, . . . , nk a − → a1n1 + · · · + aknk ϕ : k[x1, . . . , xk] − → k[w] xi − → wni IS = ker(ϕ) = xa − xb : π(a) = π(b) Ij = xa − xb : π(a) = π(b) and

  • |a| − |b|
  • ≤ j ⊂ IS

Idea: only connect some of the factorizations I0 ⊂ I1 ⊂ I2 ⊂ I3 ⊂ I4 ⊂ · · · ⊂ IS Example: S6, 9, 20 IS = x3 − y2, x4y4 − z3 ⊂ k[x, y, z] Z(244): connected components: 2

Christopher O’Neill (SDSU) Computing delta sets of affine semigroups Feb 10, 2020 23 / 27

slide-175
SLIDE 175

The delta set via commutative algebra

π : Zk

≥0

− → S = n1, . . . , nk a − → a1n1 + · · · + aknk ϕ : k[x1, . . . , xk] − → k[w] xi − → wni IS = ker(ϕ) = xa − xb : π(a) = π(b) Ij = xa − xb : π(a) = π(b) and

  • |a| − |b|
  • ≤ j ⊂ IS

Idea: only connect some of the factorizations I0 ⊂ I1 ⊂ I2 ⊂ I3 ⊂ I4 ⊂ · · · ⊂ IS Example: S6, 9, 20 IS = x3 − y2, x4y4 − z3 ⊂ k[x, y, z] Z(244): connected components: 2

Christopher O’Neill (SDSU) Computing delta sets of affine semigroups Feb 10, 2020 23 / 27

slide-176
SLIDE 176

The delta set via commutative algebra

π : Zk

≥0

− → S = n1, . . . , nk a − → a1n1 + · · · + aknk ϕ : k[x1, . . . , xk] − → k[w] xi − → wni IS = ker(ϕ) = xa − xb : π(a) = π(b) Ij = xa − xb : π(a) = π(b) and

  • |a| − |b|
  • ≤ j ⊂ IS

Idea: only connect some of the factorizations I0 ⊂ I1 ⊂ I2 ⊂ I3 ⊂ I4 ⊂ · · · ⊂ IS Example: S6, 9, 20 IS = x3 − y2, x4y4 − z3 ⊂ k[x, y, z] Z(244): connected components: 1

Christopher O’Neill (SDSU) Computing delta sets of affine semigroups Feb 10, 2020 23 / 27

slide-177
SLIDE 177

The delta set via commutative algebra

π : Zk

≥0

− → S = n1, . . . , nk a − → a1n1 + · · · + aknk ϕ : k[x1, . . . , xk] − → k[w] xi − → wni IS = ker(ϕ) = xa − xb : π(a) = π(b) Ij = xa − xb : π(a) = π(b) and

  • |a| − |b|
  • ≤ j ⊂ IS

Idea: only connect some of the factorizations I0 ⊂ I1 ⊂ I2 ⊂ I3 ⊂ I4 ⊂ · · · ⊂ IS Example: S6, 9, 20 IS = x3 − y2, x4y4 − z3 ⊂ k[x, y, z] Z(244): connected components: 1

Christopher O’Neill (SDSU) Computing delta sets of affine semigroups Feb 10, 2020 23 / 27

slide-178
SLIDE 178

The delta set via commutative algebra

π : Zk

≥0

− → S = n1, . . . , nk a − → a1n1 + · · · + aknk ϕ : k[x1, . . . , xk] − → k[w] xi − → wni IS = ker(ϕ) = xa − xb : π(a) = π(b) Ij = xa − xb : π(a) = π(b) and

  • |a| − |b|
  • ≤ j ⊂ IS

Idea: only connect some of the factorizations I0 ⊂ I1 ⊂ I2 ⊂ I3 ⊂ I4 ⊂ · · · ⊂ IS Example: S6, 9, 20 IS = x3 − y2, x4y4 − z3 ⊂ k[x, y, z] Z(244): connected components: 1

Christopher O’Neill (SDSU) Computing delta sets of affine semigroups Feb 10, 2020 23 / 27

slide-179
SLIDE 179

The delta set via commutative algebra

π : Zk

≥0

− → S = n1, . . . , nk a − → a1n1 + · · · + aknk ϕ : k[x1, . . . , xk] − → k[w] xi − → wni IS = ker(ϕ) = xa − xb : π(a) = π(b) Ij = xa − xb : π(a) = π(b) and

  • |a| − |b|
  • ≤ j ⊂ IS

Idea: only connect some of the factorizations I0 ⊂ I1 ⊂ I2 ⊂ I3 ⊂ I4 ⊂ · · · ⊂ IS Example: S6, 9, 20 IS = x3 − y2, x4y4 − z3 ⊂ k[x, y, z] Z(244): connected components: 1

Christopher O’Neill (SDSU) Computing delta sets of affine semigroups Feb 10, 2020 23 / 27

slide-180
SLIDE 180

The delta set via commutative algebra

π : Zk

≥0

− → S = n1, . . . , nk a − → a1n1 + · · · + aknk ϕ : k[x1, . . . , xk] − → k[w] xi − → wni IS = ker(ϕ) = xa − xb : π(a) = π(b) Ij = xa − xb : π(a) = π(b) and

  • |a| − |b|
  • ≤ j ⊂ IS

Idea: only connect some of the factorizations I0 ⊂ I1 ⊂ I2 ⊂ I3 ⊂ I4 ⊂ · · · ⊂ IS Example: S6, 9, 20 IS = x3 − y2, x4y4 − z3 ⊂ k[x, y, z] Z(244): connected components: 1 I0 = x11z3 − y14 I1 = I0 + x3 − y2, x8z3 − y12 I2 = I1 + x5z3 − y10 I3 = I2 + x2z3 − y8 I4 = I3 + x4y4 − z3 = I5 = I6 = · · · = IS

Christopher O’Neill (SDSU) Computing delta sets of affine semigroups Feb 10, 2020 23 / 27

slide-181
SLIDE 181

The delta set via commutative algebra

π : Zk

≥0

− → S = n1, . . . , nk a − → a1n1 + · · · + aknk ϕ : k[x1, . . . , xk] − → k[w] xi − → wni IS = ker(ϕ) = xa − xb : π(a) = π(b) Ij = xa − xb : π(a) = π(b) and

  • |a| − |b|
  • ≤ j ⊂ IS

Christopher O’Neill (SDSU) Computing delta sets of affine semigroups Feb 10, 2020 24 / 27

slide-182
SLIDE 182

The delta set via commutative algebra

π : Zk

≥0

− → S = n1, . . . , nk a − → a1n1 + · · · + aknk ϕ : k[x1, . . . , xk] − → k[w] xi − → wni IS = ker(ϕ) = xa − xb : π(a) = π(b) Ij = xa − xb : π(a) = π(b) and

  • |a| − |b|
  • ≤ j ⊂ IS

Theorem (O, 2016)

In the ascending chain I0 ⊂ I1 ⊂ I2 ⊂ · · · ⊂ IS, j ∈ ∆(S) if and only if Ij−1 Ij

Christopher O’Neill (SDSU) Computing delta sets of affine semigroups Feb 10, 2020 24 / 27

slide-183
SLIDE 183

The delta set via commutative algebra

π : Zk

≥0

− → S = n1, . . . , nk a − → a1n1 + · · · + aknk ϕ : k[x1, . . . , xk] − → k[w] xi − → wni IS = ker(ϕ) = xa − xb : π(a) = π(b) Ij = xa − xb : π(a) = π(b) and

  • |a| − |b|
  • ≤ j ⊂ IS

Theorem (O, 2016)

In the ascending chain I0 ⊂ I1 ⊂ I2 ⊂ · · · ⊂ IS, j ∈ ∆(S) if and only if Ij−1 Ij Algorithm for computing ∆(S): Compute generators for I0, I1, . . . At each step, check if Ij−1 = Ij Stop when IS reached

Christopher O’Neill (SDSU) Computing delta sets of affine semigroups Feb 10, 2020 24 / 27

slide-184
SLIDE 184

The delta set via commutative algebra

π : Zk

≥0

− → S = n1, . . . , nk a − → a1n1 + · · · + aknk ϕ : k[x1, . . . , xk] − → k[w] xi − → wni IS = ker(ϕ) = xa − xb : π(a) = π(b) Ij = xa − xb : π(a) = π(b) and

  • |a| − |b|
  • ≤ j ⊂ IS

Christopher O’Neill (SDSU) Computing delta sets of affine semigroups Feb 10, 2020 25 / 27

slide-185
SLIDE 185

The delta set via commutative algebra

π : Zk

≥0

− → S = n1, . . . , nk a − → a1n1 + · · · + aknk ϕ : k[x1, . . . , xk] − → k[w] xi − → wni IS = ker(ϕ) = xa − xb : π(a) = π(b) Ij = xa − xb : π(a) = π(b) and

  • |a| − |b|
  • ≤ j ⊂ IS

Ihom: homogenization of IS xa − xb ∈ IS − → xa − t|a|−|b|xb ∈ Ihom

Christopher O’Neill (SDSU) Computing delta sets of affine semigroups Feb 10, 2020 25 / 27

slide-186
SLIDE 186

The delta set via commutative algebra

π : Zk

≥0

− → S = n1, . . . , nk a − → a1n1 + · · · + aknk ϕ : k[x1, . . . , xk] − → k[w] xi − → wni IS = ker(ϕ) = xa − xb : π(a) = π(b) Ij = xa − xb : π(a) = π(b) and

  • |a| − |b|
  • ≤ j ⊂ IS

Ihom: homogenization of IS xa − xb ∈ IS − → xa − t|a|−|b|xb ∈ Ihom Example: S6, 9, 20 IS = x3 − y2, xy6 − z3 ⊂ k[x, y, z]

Christopher O’Neill (SDSU) Computing delta sets of affine semigroups Feb 10, 2020 25 / 27

slide-187
SLIDE 187

The delta set via commutative algebra

π : Zk

≥0

− → S = n1, . . . , nk a − → a1n1 + · · · + aknk ϕ : k[x1, . . . , xk] − → k[w] xi − → wni IS = ker(ϕ) = xa − xb : π(a) = π(b) Ij = xa − xb : π(a) = π(b) and

  • |a| − |b|
  • ≤ j ⊂ IS

Ihom: homogenization of IS xa − xb ∈ IS − → xa − t|a|−|b|xb ∈ Ihom Example: S6, 9, 20 IS = x3 − y2, xy6 − z3 ⊂ k[x, y, z] Lex Gr¨

  • bner basis for Ihom:

Ihom = x11z3 − y14, x3 − ty2, x8z3 − ty12, t2x5z3 − y10, t3x2z3 − y8, xy6 − t4z3

Christopher O’Neill (SDSU) Computing delta sets of affine semigroups Feb 10, 2020 25 / 27

slide-188
SLIDE 188

The delta set via commutative algebra

π : Zk

≥0

− → S = n1, . . . , nk a − → a1n1 + · · · + aknk ϕ : k[x1, . . . , xk] − → k[w] xi − → wni IS = ker(ϕ) = xa − xb : π(a) = π(b) Ij = xa − xb : π(a) = π(b) and

  • |a| − |b|
  • ≤ j ⊂ IS

Ihom: homogenization of IS xa − xb ∈ IS − → xa − t|a|−|b|xb ∈ Ihom Example: S6, 9, 20 IS = x3 − y2, xy6 − z3 ⊂ k[x, y, z] Lex Gr¨

  • bner basis for Ihom:

Ihom = x11z3 − y14, x3 − ty2, x8z3 − ty12, t2x5z3 − y10, t3x2z3 − y8, xy6 − t4z3 I0 = x11z3 − y14 I1 = I0 + x3 − y2, x8z3 − y12 I2 = I1 + x5z3 − y10 I3 = I2 + x2z3 − y8 I4 = I3 + xy6 − z3

Christopher O’Neill (SDSU) Computing delta sets of affine semigroups Feb 10, 2020 25 / 27

slide-189
SLIDE 189

The delta set via commutative algebra

π : Zk

≥0

− → S = n1, . . . , nk a − → a1n1 + · · · + aknk ϕ : k[x1, . . . , xk] − → k[w] xi − → wni IS = ker(ϕ) = xa − xb : π(a) = π(b) Ij = xa − xb : π(a) = π(b) and

  • |a| − |b|
  • ≤ j ⊂ IS

Christopher O’Neill (SDSU) Computing delta sets of affine semigroups Feb 10, 2020 26 / 27

slide-190
SLIDE 190

The delta set via commutative algebra

π : Zk

≥0

− → S = n1, . . . , nk a − → a1n1 + · · · + aknk ϕ : k[x1, . . . , xk] − → k[w] xi − → wni IS = ker(ϕ) = xa − xb : π(a) = π(b) Ij = xa − xb : π(a) = π(b) and

  • |a| − |b|
  • ≤ j ⊂ IS

Algorithm to compute ∆(S) (Garc´ ıa-S´ anchez–O–Webb, 2018)

Homogenize the ideal IS with a new variable t Compute a reduced lex Gr¨

  • bner basis G with t < xi

∆(S) = {d : tdxa − xb ∈ G}

Christopher O’Neill (SDSU) Computing delta sets of affine semigroups Feb 10, 2020 26 / 27

slide-191
SLIDE 191

The delta set via commutative algebra

π : Zk

≥0

− → S = n1, . . . , nk a − → a1n1 + · · · + aknk ϕ : k[x1, . . . , xk] − → k[w] xi − → wni IS = ker(ϕ) = xa − xb : π(a) = π(b) Ij = xa − xb : π(a) = π(b) and

  • |a| − |b|
  • ≤ j ⊂ IS

Algorithm to compute ∆(S) (Garc´ ıa-S´ anchez–O–Webb, 2018)

Homogenize the ideal IS with a new variable t Compute a reduced lex Gr¨

  • bner basis G with t < xi

∆(S) = {d : tdxa − xb ∈ G}

S ∆(S) Manual Dynamic Algebraic 100, 121, 142, 163, 284 {21} Days 0m 3.6s < 10 ms 1001, 1211, 1421, 1631, 2841 {10, 20, 30} Days 1m 56s < 10 ms

Christopher O’Neill (SDSU) Computing delta sets of affine semigroups Feb 10, 2020 26 / 27

slide-192
SLIDE 192

The delta set via commutative algebra

π : Zk

≥0

− → S = n1, . . . , nk a − → a1n1 + · · · + aknk ϕ : k[x1, . . . , xk] − → k[w] xi − → wni IS = ker(ϕ) = xa − xb : π(a) = π(b) Ij = xa − xb : π(a) = π(b) and

  • |a| − |b|
  • ≤ j ⊂ IS

Algorithm to compute ∆(S) (Garc´ ıa-S´ anchez–O–Webb, 2018)

Homogenize the ideal IS with a new variable t Compute a reduced lex Gr¨

  • bner basis G with t < xi

∆(S) = {d : tdxa − xb ∈ G}

S ∆(S) Manual Dynamic Algebraic 100, 121, 142, 163, 284 {21} Days 0m 3.6s < 10 ms 1001, 1211, 1421, 1631, 2841 {10, 20, 30} Days 1m 56s < 10 ms 550, 1060, 1600, 1781, 4126, 4139, 4407, 5167, 6073, 6079, 6169, 7097, 7602, 8782, 8872    1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 19    Years Days < 1 min

Christopher O’Neill (SDSU) Computing delta sets of affine semigroups Feb 10, 2020 26 / 27

slide-193
SLIDE 193

References

  • T. Barron, C. O’Neill, R. Pelayo (2015)

On the computation of delta sets and ω-primality in numerical monoids. preprint.

  • J. Garc´

ıa-Garc´ ıa, M. Moreno-Fr´ ıas, A. Vigneron-Tenorio (2014) Computation of delta sets of numerical monoids. preprint.

  • P. Garc´

ıa-S´ anchez, C. O’Neill, G. Webb (2017) On the computation of factorization invariants for affine semigroups. preprint.

  • M. Delgado, P. Garc´

ıa-S´ anchez, J. Morais GAP Numerical Semigroups Package http://www.gap-system.org/Packages/numericalsgps.html.

Christopher O’Neill (SDSU) Computing delta sets of affine semigroups Feb 10, 2020 27 / 27

slide-194
SLIDE 194

References

  • T. Barron, C. O’Neill, R. Pelayo (2015)

On the computation of delta sets and ω-primality in numerical monoids. preprint.

  • J. Garc´

ıa-Garc´ ıa, M. Moreno-Fr´ ıas, A. Vigneron-Tenorio (2014) Computation of delta sets of numerical monoids. preprint.

  • P. Garc´

ıa-S´ anchez, C. O’Neill, G. Webb (2017) On the computation of factorization invariants for affine semigroups. preprint.

  • M. Delgado, P. Garc´

ıa-S´ anchez, J. Morais GAP Numerical Semigroups Package http://www.gap-system.org/Packages/numericalsgps.html. Thanks!

Christopher O’Neill (SDSU) Computing delta sets of affine semigroups Feb 10, 2020 27 / 27