[PPT] - Numerical semigroups: a sales pitch Christopher ONeill San Diego PowerPoint Presentation

SLIDE 1

Numerical semigroups: a sales pitch

Christopher O’Neill

San Diego State University cdoneill@sdsu.edu Slides available: http://www.tinyurl.com/JMM2020-FURM

January 17, 2020

Christopher O’Neill (SDSU) Numerical semigroup: a sales pitch January 17, 2020 1 / 14

SLIDE 2

Numerical semigroups

Definition

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

Christopher O’Neill (SDSU) Numerical semigroup: a sales pitch January 17, 2020 2 / 14

SLIDE 3

Numerical 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) Numerical semigroup: a sales pitch January 17, 2020 2 / 14

SLIDE 4

Numerical 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) Numerical semigroup: a sales pitch January 17, 2020 2 / 14

SLIDE 5

Numerical 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) Numerical semigroup: a sales pitch January 17, 2020 2 / 14

SLIDE 6

Numerical 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) Numerical semigroup: a sales pitch January 17, 2020 2 / 14

SLIDE 7

Numerical 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) Numerical semigroup: a sales pitch January 17, 2020 2 / 14

SLIDE 8

Numerical 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) Numerical semigroup: a sales pitch January 17, 2020 2 / 14

SLIDE 9

Extremal factorization length

Fix a numerical semigroup S = n1, . . . , nk and an element n ∈ S.

Christopher O’Neill (SDSU) Numerical semigroup: a sales pitch January 17, 2020 3 / 14

SLIDE 10

Extremal factorization length

Fix a numerical semigroup S = n1, . . . , nk and an element n ∈ S. A factorization a = (a1, . . . , ak) ∈ Zk

≥0 of n

n = a1n1 + · · · + aknk has length |a| = a1 + · · · + ak.

Christopher O’Neill (SDSU) Numerical semigroup: a sales pitch January 17, 2020 3 / 14

SLIDE 11

Extremal factorization length

Fix a numerical semigroup S = n1, . . . , nk and an element n ∈ S. A factorization a = (a1, . . . , ak) ∈ Zk

≥0 of n

n = a1n1 + · · · + aknk has length |a| = a1 + · · · + ak.

Example

All factorizations of 60 ∈ 6, 9, 20: (10, 0, 0), (7, 2, 0), (4, 4, 0), (1, 6, 0), (0, 0, 3)

Christopher O’Neill (SDSU) Numerical semigroup: a sales pitch January 17, 2020 3 / 14

SLIDE 12

Extremal factorization length

Fix a numerical semigroup S = n1, . . . , nk and an element n ∈ S. A factorization a = (a1, . . . , ak) ∈ Zk

≥0 of n

n = a1n1 + · · · + aknk has length |a| = a1 + · · · + ak.

Example

All factorizations of 60 ∈ 6, 9, 20: (10, 0, 0), (7, 2, 0), (4, 4, 0), (1, 6, 0), (0, 0, 3) Lengths: 3, 7, 8, 9, 10.

Christopher O’Neill (SDSU) Numerical semigroup: a sales pitch January 17, 2020 3 / 14

SLIDE 13

Extremal factorization length

Fix a numerical semigroup S = n1, . . . , nk and an element n ∈ S. A factorization a = (a1, . . . , ak) ∈ Zk

≥0 of n

n = a1n1 + · · · + aknk has length |a| = a1 + · · · + ak.

Example

All factorizations of 60 ∈ 6, 9, 20: (10, 0, 0), (7, 2, 0), (4, 4, 0), (1, 6, 0), (0, 0, 3) Lengths: 3, 7, 8, 9, 10. All factorizations of 1000001:

Christopher O’Neill (SDSU) Numerical semigroup: a sales pitch January 17, 2020 3 / 14

SLIDE 14

Extremal factorization length

Fix a numerical semigroup S = n1, . . . , nk and an element n ∈ S. A factorization a = (a1, . . . , ak) ∈ Zk

≥0 of n

n = a1n1 + · · · + aknk has length |a| = a1 + · · · + ak.

Example

All factorizations of 60 ∈ 6, 9, 20: (10, 0, 0), (7, 2, 0), (4, 4, 0), (1, 6, 0), (0, 0, 3) Lengths: 3, 7, 8, 9, 10. All factorizations of 1000001:

shortest

, . . . ,

longest

Christopher O’Neill (SDSU) Numerical semigroup: a sales pitch January 17, 2020 3 / 14

SLIDE 15

Extremal factorization length

Fix a numerical semigroup S = n1, . . . , nk and an element n ∈ S. A factorization a = (a1, . . . , ak) ∈ Zk

≥0 of n

n = a1n1 + · · · + aknk has length |a| = a1 + · · · + ak.

Example

All factorizations of 60 ∈ 6, 9, 20: (10, 0, 0), (7, 2, 0), (4, 4, 0), (1, 6, 0), (0, 0, 3) Lengths: 3, 7, 8, 9, 10. All factorizations of 1000001: (2, 1, 49999)

shortest

, . . . ,

longest

Christopher O’Neill (SDSU) Numerical semigroup: a sales pitch January 17, 2020 3 / 14

SLIDE 16

Extremal factorization length

Fix a numerical semigroup S = n1, . . . , nk and an element n ∈ S. A factorization a = (a1, . . . , ak) ∈ Zk

≥0 of n

n = a1n1 + · · · + aknk has length |a| = a1 + · · · + ak.

Example

All factorizations of 60 ∈ 6, 9, 20: (10, 0, 0), (7, 2, 0), (4, 4, 0), (1, 6, 0), (0, 0, 3) Lengths: 3, 7, 8, 9, 10. All factorizations of 1000001: (2, 1, 49999)

shortest

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

longest

Christopher O’Neill (SDSU) Numerical semigroup: a sales pitch January 17, 2020 3 / 14

SLIDE 17

Extremal factorization length

Let S = n1, . . . , nk. For n ∈ S, let L(n) = {a1 + · · · + ak : n = a1n1 + · · · + aknk} denotes the length set of n

Christopher O’Neill (SDSU) Numerical semigroup: a sales pitch January 17, 2020 4 / 14

SLIDE 18

Extremal factorization length

Let S = n1, . . . , nk. For n ∈ S, let L(n) = {a1 + · · · + ak : n = a1n1 + · · · + aknk} denotes the length set of n, and M(n) = max L(n) and m(n) = min L(n) denote the maximum and minimum factorization lengths of n.

Christopher O’Neill (SDSU) Numerical semigroup: a sales pitch January 17, 2020 4 / 14

SLIDE 19

Extremal factorization length

Let S = n1, . . . , nk. For n ∈ S, let L(n) = {a1 + · · · + ak : n = a1n1 + · · · + aknk} denotes the length set of n, and M(n) = max L(n) and m(n) = min L(n) denote the maximum and minimum factorization lengths of n.

Observations

Max length factorization: lots of small generators Min length factorization: lots of large generators

Christopher O’Neill (SDSU) Numerical semigroup: a sales pitch January 17, 2020 4 / 14

SLIDE 20

Extremal factorization length

Let S = n1, . . . , nk. For n ∈ S, let L(n) = {a1 + · · · + ak : n = a1n1 + · · · + aknk} denotes the length set of n, and M(n) = max L(n) and m(n) = min L(n) denote the maximum and minimum factorization lengths of n.

Observations

Max length factorization: lots of small generators Min length factorization: lots of large generators

Example

S = 5, 16, 17, 18, 19:

Christopher O’Neill (SDSU) Numerical semigroup: a sales pitch January 17, 2020 4 / 14

SLIDE 21

Extremal factorization length

Let S = n1, . . . , nk. For n ∈ S, let L(n) = {a1 + · · · + ak : n = a1n1 + · · · + aknk} denotes the length set of n, and M(n) = max L(n) and m(n) = min L(n) denote the maximum and minimum factorization lengths of n.

Observations

Max length factorization: lots of small generators Min length factorization: lots of large generators

Example

S = 5, 16, 17, 18, 19: m(82) = 5 with 82 = 3(16) + 2(17) m(462) = 25 with 462 = 3(16) + 2(17) + 20(19)

Christopher O’Neill (SDSU) Numerical semigroup: a sales pitch January 17, 2020 4 / 14

SLIDE 22

Extremal factorization length

Let S = n1, . . . , nk. For n ∈ S, let M(n) = max L(n) and m(n) = min L(n).

10 20 30 40 50 60 70 80 2 4 6 8 10 12

Christopher O’Neill (SDSU) Numerical semigroup: a sales pitch January 17, 2020 5 / 14

SLIDE 23

Extremal factorization length

Let S = n1, . . . , nk. For n ∈ S, let M(n) = max L(n) and m(n) = min L(n). Example: max length in S = 6, 9, 20

10 20 30 40 50 60 70 80 2 4 6 8 10 12

Christopher O’Neill (SDSU) Numerical semigroup: a sales pitch January 17, 2020 5 / 14

SLIDE 24

Extremal factorization length

Let S = n1, . . . , nk. For n ∈ S, let M(n) = max L(n) and m(n) = min L(n). Example: max length in S = 6, 9, 20

10 20 30 40 50 60 70 80 2 4 6 8 10 12

M(n) : S → N

Christopher O’Neill (SDSU) Numerical semigroup: a sales pitch January 17, 2020 5 / 14

SLIDE 25

Extremal factorization length

Let S = n1, . . . , nk. For n ∈ S, let M(n) = max L(n) and m(n) = min L(n). Example: max length in S = 6, 9, 20

10 20 30 40 50 60 70 80 2 4 6 8 10 12

M(n) : S → N For n ≥ 49, M(n) =

                      

1 6n

if n ≡ 0 mod 6

1 6n − 31 6

if n ≡ 1 mod 6

1 6n − 7 3

if n ≡ 2 mod 6

1 6n − 1 2

if n ≡ 3 mod 6

1 6n − 14 3

if n ≡ 4 mod 6

1 6n − 17 6

if n ≡ 5 mod 6

Christopher O’Neill (SDSU) Numerical semigroup: a sales pitch January 17, 2020 5 / 14

SLIDE 26

Extremal factorization length

Let S = n1, . . . , nk. For n ∈ S, let M(n) = max L(n) and m(n) = min L(n). Example: max length in S = 6, 9, 20

10 20 30 40 50 60 70 80 2 4 6 8 10 12

M(n) : S → N For n ≥ 49, M(n) =

                      

1 6n

if n ≡ 0 mod 6

1 6n − 31 6

if n ≡ 1 mod 6

1 6n − 7 3

if n ≡ 2 mod 6

1 6n − 1 2

if n ≡ 3 mod 6

1 6n − 14 3

if n ≡ 4 mod 6

1 6n − 17 6

if n ≡ 5 mod 6 We say M(n) is (eventually) quasilinear: linear with periodic coefficients.

Christopher O’Neill (SDSU) Numerical semigroup: a sales pitch January 17, 2020 5 / 14

SLIDE 27

Extremal factorization length

Let S = n1, . . . , nk. For n ∈ S, let M(n) = max L(n) and m(n) = min L(n). Example: max length in S = 6, 9, 20

10 20 30 40 50 60 70 80 2 4 6 8 10 12

M(n) : S → N For n ≥ 49, M(n) =

                      

1 6n

if n ≡ 0 mod 6

1 6n − 31 6

if n ≡ 1 mod 6

1 6n − 7 3

if n ≡ 2 mod 6

1 6n − 1 2

if n ≡ 3 mod 6

1 6n − 14 3

if n ≡ 4 mod 6

1 6n − 17 6

if n ≡ 5 mod 6 We say M(n) is (eventually) quasilinear: linear with periodic coefficients. M(n) = 1

6n + a0(n)

for some periodic function a0(n)

Christopher O’Neill (SDSU) Numerical semigroup: a sales pitch January 17, 2020 5 / 14

SLIDE 28

Extremal factorization length

Let S = n1, . . . , nk. For n ∈ S, let M(n) = max L(n) and m(n) = min L(n).

20 40 60 80 100 1 2 3 4 5 6

Christopher O’Neill (SDSU) Numerical semigroup: a sales pitch January 17, 2020 6 / 14

SLIDE 29

Extremal factorization length

Let S = n1, . . . , nk. For n ∈ S, let M(n) = max L(n) and m(n) = min L(n). Example: min length in S = 5, 16, 17, 18, 19

20 40 60 80 100 1 2 3 4 5 6

m(n) : S → N

Christopher O’Neill (SDSU) Numerical semigroup: a sales pitch January 17, 2020 6 / 14

SLIDE 30

Extremal factorization length

Let S = n1, . . . , nk. For n ∈ S, let M(n) = max L(n) and m(n) = min L(n). Example: min length in S = 5, 16, 17, 18, 19

20 40 60 80 100 1 2 3 4 5 6

m(n) : S → N Conclusion: m(n) is quasilinear for n ≥ 64, with period 19.

Christopher O’Neill (SDSU) Numerical semigroup: a sales pitch January 17, 2020 6 / 14

SLIDE 31

Extremal factorization length

Let S = n1, . . . , nk. For n ∈ S, M(n) = max L(n) and m(n) = min L(n).

Christopher O’Neill (SDSU) Numerical semigroup: a sales pitch January 17, 2020 7 / 14

SLIDE 32

Extremal factorization length

Let S = n1, . . . , nk. For n ∈ S, M(n) = max L(n) and m(n) = min L(n).

Theorem

Let S = n1, . . . , nk. For n ≫ 0 (i.e., for n sufficiently large), M(n + n1) = 1 + M(n) m(n + nk) = 1 + m(n)

Christopher O’Neill (SDSU) Numerical semigroup: a sales pitch January 17, 2020 7 / 14

SLIDE 33

Extremal factorization length

Let S = n1, . . . , nk. For n ∈ S, M(n) = max L(n) and m(n) = min L(n).

Theorem

Let S = n1, . . . , nk. For n ≫ 0 (i.e., for n sufficiently large), M(n + n1) = 1 + M(n) m(n + nk) = 1 + m(n) Equivalently, M(n), m(n) are eventually quasilinear: M(n) =

1 n1 n + a0(n)

m(n) =

1 nk n + b0(n)

for periodic functions a0(n), b0(n).

Christopher O’Neill (SDSU) Numerical semigroup: a sales pitch January 17, 2020 7 / 14

SLIDE 34

Extremal factorization length

Let S = n1, . . . , nk. For n ∈ S, M(n) = max L(n) and m(n) = min L(n).

Theorem

Let S = n1, . . . , nk. For n ≫ 0 (i.e., for n sufficiently large), M(n + n1) = 1 + M(n) m(n + nk) = 1 + m(n) Equivalently, M(n), m(n) are eventually quasilinear: M(n) =

1 n1 n + a0(n)

m(n) =

1 nk n + b0(n)

for periodic functions a0(n), b0(n). M(n) =

      

1 n1 n +

if n ≡ 0 mod n1

1 n1 n +

if n ≡ 1 mod n1 · · ·

Christopher O’Neill (SDSU) Numerical semigroup: a sales pitch January 17, 2020 7 / 14

SLIDE 35

Plenty more where that came from!

Let S = n1, . . . , nk ⊂ Z≥0. For n ≫ 0, we have

Christopher O’Neill (SDSU) Numerical semigroup: a sales pitch January 17, 2020 8 / 14

SLIDE 36

Plenty more where that came from!

Let S = n1, . . . , nk ⊂ Z≥0. For n ≫ 0, we have M(n) = 1

n1 n +

periodic

func’n

and

m(n) = 1

nk n +

periodic

func’n

T. Barron, C. O’Neill, and R. Pelayo,

On the set of elasticities in numerical monoids, Semigroup Forum 94 (2017), no. 1, 37–50. [arXiv:1409.3425]

|L(n)| = nk−n1

δn1nk n +

periodic

func’n

C. O’Neill,

On factorization invariants and Hilbert functions,

J. Pure and Applied Algebra 221 (2017), no. 12, 3069–3088. [arXiv:1503.08351]

the delta set invariant is periodic in n: ∆(n) = ∆(n + n1nk)

S. Chapman, R. Hoyer, and N. Kaplan,

Delta sets of numerical monoids are eventually periodic, Aequationes mathematicae 77 3 (2009) 273–279. [doi]

the ω-primality invariant ω(n) = 1

n1 n +

periodic

func’n

C. O’Neill and R. Pelayo,

On the linearity of ω-primality in numerical monoids,

J. Pure and Applied Algebra 218 (2014) 1620–1627. [arXiv:1309.7476]

the catenary degree invariant c(n) is periodic in n

S. Chapman, M. Corrales, A. Miller, C. Miller, and D. Patel,

The catenary and tame degrees on a numerical monoid are eventually periodic,

J. Aust. Math. Soc. 97 (2014), no. 3, 289–300. [doi]

Christopher O’Neill (SDSU) Numerical semigroup: a sales pitch January 17, 2020 8 / 14

SLIDE 37

Plenty more where that came from!

Let S = n1, . . . , nk ⊂ Z≥0. For n ≫ 0, we have M(n) = 1

n1 n +

periodic

func’n

and

m(n) = 1

nk n +

periodic

func’n

T. Barron, C. O’Neill, and R. Pelayo,

On the set of elasticities in numerical monoids, Semigroup Forum 94 (2017), no. 1, 37–50. [arXiv:1409.3425]

|L(n)| = nk−n1

δn1nk n +

periodic

func’n

C. O’Neill,

On factorization invariants and Hilbert functions,

J. Pure and Applied Algebra 221 (2017), no. 12, 3069–3088. [arXiv:1503.08351]

the delta set invariant is periodic in n: ∆(n) = ∆(n + n1nk)

S. Chapman, R. Hoyer, and N. Kaplan,

Delta sets of numerical monoids are eventually periodic, Aequationes mathematicae 77 3 (2009) 273–279. [doi]

the ω-primality invariant ω(n) = 1

n1 n +

periodic

func’n

C. O’Neill and R. Pelayo,

On the linearity of ω-primality in numerical monoids,

J. Pure and Applied Algebra 218 (2014) 1620–1627. [arXiv:1309.7476]

the catenary degree invariant c(n) is periodic in n

S. Chapman, M. Corrales, A. Miller, C. Miller, and D. Patel,

The catenary and tame degrees on a numerical monoid are eventually periodic,

J. Aust. Math. Soc. 97 (2014), no. 3, 289–300. [doi]

Christopher O’Neill (SDSU) Numerical semigroup: a sales pitch January 17, 2020 8 / 14

SLIDE 38

Parametrized families of numerical semigroups

Fix r1 < · · · < rk ∈ Z≥1, and consider the parametrized family j

Mj = j, j + r1, . . . , j + rk.

Christopher O’Neill (SDSU) Numerical semigroup: a sales pitch January 17, 2020 9 / 14

SLIDE 39

Parametrized families of numerical semigroups

Fix r1 < · · · < rk ∈ Z≥1, and consider the parametrized family j

Mj = j, j + r1, . . . , j + rk.

Frobenius number F(Mj) = max(Z≥0 \ Mj): maximum “gap” of Mj.

Christopher O’Neill (SDSU) Numerical semigroup: a sales pitch January 17, 2020 9 / 14

SLIDE 40

Parametrized families of numerical semigroups

Fix r1 < · · · < rk ∈ Z≥1, and consider the parametrized family j

Mj = j, j + r1, . . . , j + rk.

Frobenius number F(Mj) = max(Z≥0 \ Mj): maximum “gap” of Mj.

Theorem

For j ≫ 0, we have F(Mj) = 1

rk j2 +

periodic

func’n

j + periodic

func’n

.

Christopher O’Neill (SDSU) Numerical semigroup: a sales pitch January 17, 2020 9 / 14

SLIDE 41

Parametrized families of numerical semigroups

Fix r1 < · · · < rk ∈ Z≥1, and consider the parametrized family j

Mj = j, j + r1, . . . , j + rk.

Frobenius number F(Mj) = max(Z≥0 \ Mj): maximum “gap” of Mj.

Theorem

For j ≫ 0, we have F(Mj) = 1

rk j2 +

periodic

func’n

j + periodic

func’n

.

Mj = j, j + 6, j + 9, j + 20: F(Mj) = 1

20j2 + · · · for j ≥ 44.

Christopher O’Neill (SDSU) Numerical semigroup: a sales pitch January 17, 2020 9 / 14

SLIDE 42

Parametrized families of numerical semigroups

Fix r1 < · · · < rk ∈ Z≥1, and consider the parametrized family j

Mj = j, j + r1, . . . , j + rk.

Frobenius number F(Mj) = max(Z≥0 \ Mj): maximum “gap” of Mj.

Theorem

For j ≫ 0, we have F(Mj) = 1

rk j2 +

periodic

func’n

j + periodic

func’n

.

Mj = j, j + 6, j + 9, j + 20: F(Mj) = 1

20j2 + · · · for j ≥ 44.

20 40 60 80 100 120 200 400 600 800 1000 1200 1400

Christopher O’Neill (SDSU) Numerical semigroup: a sales pitch January 17, 2020 9 / 14

SLIDE 43

Parametrized families of numerical semigroups

Fix r1 < · · · < rk ∈ Z≥1, and consider the parametrized family j

Mj = j, j + r1, . . . , j + rk.

Frobenius number F(Mj) = max(Z≥0 \ Mj): maximum “gap” of Mj.

Theorem

For j ≫ 0, we have F(Mj) = 1

rk j2 +

periodic

func’n

j + periodic

func’n

.

Mj = j, j + 6, j + 9, j + 20: F(Mj) = 1

20j2 + · · · for j ≥ 44.

20 40 60 80 100 120 200 400 600 800 1000 1200 1400

Christopher O’Neill (SDSU) Numerical semigroup: a sales pitch January 17, 2020 9 / 14

SLIDE 44

Parametrized families of numerical semigroups

Within a parametrized family j Mj = f1(j), . . . , fk(j), Frobenius number F(Mj) genus g(Mj) = |Z≥0 \ Mj| type t(Mj) delta sets ∆(Mj) catenary degree c(Mj) Betti numbers β(Mj) are quasipolynomials in j ≫ 0

Christopher O’Neill (SDSU) Numerical semigroup: a sales pitch January 17, 2020 10 / 14

SLIDE 45

Parametrized families of numerical semigroups

Within a parametrized family j Mj = f1(j), . . . , fk(j), Frobenius number F(Mj) genus g(Mj) = |Z≥0 \ Mj| type t(Mj) delta sets ∆(Mj) catenary degree c(Mj) Betti numbers β(Mj) are quasipolynomials in j ≫ 0 for shifted families Mj = j, j +r1, . . . , j +rk

S. Chapman, N. Kaplan, T. Lemburg, A. Niles, and C. Zlogar,

Shifts of generators and delta sets of numerical monoids, International Journal of Algebra and Computation 24 (2014), no. 5, 655–669. [doi]

C. O’Neill and R. Pelayo,

Ap´ ery sets of shifted numerical monoids, Advances in Applied Mathematics 97 (2018), 27–35. [arXiv:1708.09527]

R. Conaway, F. Gotti, J. Horton, C. O’Neill, R. Pelayo, M. Williams, and B.Wissman,

Minimal presentations of shifted numerical monoids, Int’l Journal of Algebra and Computation 28 (2018), no. 1, 53–68. [arXiv:1701.08555]

and more generally when f1(j), . . . , fk(j) are polynomials

B. Shen,

The parametric Frobenius problem and parametric exclusion,

preprint. [arXiv:1510.01349]
F. Kerstetter and C. O’Neill,

On parametrized families of numerical semigroups,

preprint. [arXiv:1909.04281]
T. Bogart, J. Goodrick, and K. Woods,

Periodic behavior in families of numerical and affine semigroups via parametric Presburger arithmetic,

preprint. [arXiv:1911.09136]

Christopher O’Neill (SDSU) Numerical semigroup: a sales pitch January 17, 2020 10 / 14

SLIDE 46

Parametrized families of numerical semigroups

Within a parametrized family j Mj = f1(j), . . . , fk(j), Frobenius number F(Mj) genus g(Mj) = |Z≥0 \ Mj| type t(Mj) delta sets ∆(Mj) catenary degree c(Mj) Betti numbers β(Mj) are quasipolynomials in j ≫ 0 for shifted families Mj = j, j +r1, . . . , j +rk

S. Chapman, N. Kaplan, T. Lemburg, A. Niles, and C. Zlogar,

Shifts of generators and delta sets of numerical monoids, International Journal of Algebra and Computation 24 (2014), no. 5, 655–669. [doi]

C. O’Neill and R. Pelayo,

Ap´ ery sets of shifted numerical monoids, Advances in Applied Mathematics 97 (2018), 27–35. [arXiv:1708.09527]

R. Conaway, F. Gotti, J. Horton, C. O’Neill, R. Pelayo, M. Williams, and B.Wissman,

Minimal presentations of shifted numerical monoids, Int’l Journal of Algebra and Computation 28 (2018), no. 1, 53–68. [arXiv:1701.08555]

and more generally when f1(j), . . . , fk(j) are polynomials

B. Shen,

The parametric Frobenius problem and parametric exclusion,

preprint. [arXiv:1510.01349]
F. Kerstetter and C. O’Neill,

On parametrized families of numerical semigroups,

preprint. [arXiv:1909.04281]
T. Bogart, J. Goodrick, and K. Woods,

Periodic behavior in families of numerical and affine semigroups via parametric Presburger arithmetic,

preprint. [arXiv:1911.09136]

Christopher O’Neill (SDSU) Numerical semigroup: a sales pitch January 17, 2020 10 / 14

SLIDE 47

A couple of long-standing (hard) conjectures

Fix a numerical semigroup S ⊂ Z≥0.

Christopher O’Neill (SDSU) Numerical semigroup: a sales pitch January 17, 2020 11 / 14

SLIDE 48

A couple of long-standing (hard) conjectures

Fix a numerical semigroup S ⊂ Z≥0. Genus g = g(S) = |Z≥0 \ S|: number of “gaps” of S.

Christopher O’Neill (SDSU) Numerical semigroup: a sales pitch January 17, 2020 11 / 14

SLIDE 49

A couple of long-standing (hard) conjectures

Fix a numerical semigroup S ⊂ Z≥0. Genus g = g(S) = |Z≥0 \ S|: number of “gaps” of S. ng = # of numerical semigroups with genus g.

Christopher O’Neill (SDSU) Numerical semigroup: a sales pitch January 17, 2020 11 / 14

SLIDE 50

A couple of long-standing (hard) conjectures

Fix a numerical semigroup S ⊂ Z≥0. Genus g = g(S) = |Z≥0 \ S|: number of “gaps” of S. ng = # of numerical semigroups with genus g. Example: n3 = 4 2, 7 = {0, 2, 4, 6, 7, 8, . . .} 3, 4 = {0, 3, 4, 6, 7, 8, . . .} 3, 5, 7 = {0, 3, 5, 6, 7, 8, . . .} 4, 5, 6, 7 = {0, 4, 5, 6, 7, 8, . . .}

Christopher O’Neill (SDSU) Numerical semigroup: a sales pitch January 17, 2020 11 / 14

SLIDE 51

A couple of long-standing (hard) conjectures

Fix a numerical semigroup S ⊂ Z≥0. Genus g = g(S) = |Z≥0 \ S|: number of “gaps” of S. ng = # of numerical semigroups with genus g. Example: n3 = 4 2, 7 = {0, 2, 4, 6, 7, 8, . . .} 3, 4 = {0, 3, 4, 6, 7, 8, . . .} 3, 5, 7 = {0, 3, 5, 6, 7, 8, . . .} 4, 5, 6, 7 = {0, 4, 5, 6, 7, 8, . . .} Suspected: ng ≥ ng−1 + ng−2 for all g (verified for g ≤ 60)

Christopher O’Neill (SDSU) Numerical semigroup: a sales pitch January 17, 2020 11 / 14

SLIDE 52

A couple of long-standing (hard) conjectures

Fix a numerical semigroup S ⊂ Z≥0. Genus g = g(S) = |Z≥0 \ S|: number of “gaps” of S. ng = # of numerical semigroups with genus g. Example: n3 = 4 2, 7 = {0, 2, 4, 6, 7, 8, . . .} 3, 4 = {0, 3, 4, 6, 7, 8, . . .} 3, 5, 7 = {0, 3, 5, 6, 7, 8, . . .} 4, 5, 6, 7 = {0, 4, 5, 6, 7, 8, . . .} Suspected: ng ≥ ng−1 + ng−2 for all g (verified for g ≤ 60) Known: limg→∞

ng+1 ng

= (the golden ratio)

Christopher O’Neill (SDSU) Numerical semigroup: a sales pitch January 17, 2020 11 / 14

SLIDE 53

A couple of long-standing (hard) conjectures

Fix a numerical semigroup S ⊂ Z≥0. Genus g = g(S) = |Z≥0 \ S|: number of “gaps” of S. ng = # of numerical semigroups with genus g. Example: n3 = 4 2, 7 = {0, 2, 4, 6, 7, 8, . . .} 3, 4 = {0, 3, 4, 6, 7, 8, . . .} 3, 5, 7 = {0, 3, 5, 6, 7, 8, . . .} 4, 5, 6, 7 = {0, 4, 5, 6, 7, 8, . . .} Suspected: ng ≥ ng−1 + ng−2 for all g (verified for g ≤ 60) Known: limg→∞

ng+1 ng

= (the golden ratio)

Conjecture (Bras-Amoros, 2008)

For all g, we have ng ≥ ng−1.

Christopher O’Neill (SDSU) Numerical semigroup: a sales pitch January 17, 2020 11 / 14

SLIDE 54

A couple of long-standing (hard) conjectures

Fix a numerical semigroup S ⊂ Z≥0. Genus g = g(S) = |Z≥0 \ S|: number of “gaps” of S. ng = # of numerical semigroups with genus g. Example: n3 = 4 2, 7 = {0, 2, 4, 6, 7, 8, . . .} 3, 4 = {0, 3, 4, 6, 7, 8, . . .} 3, 5, 7 = {0, 3, 5, 6, 7, 8, . . .} 4, 5, 6, 7 = {0, 4, 5, 6, 7, 8, . . .} Suspected: ng ≥ ng−1 + ng−2 for all g (verified for g ≤ 60) Known: limg→∞

ng+1 ng

= (the golden ratio)

Conjecture (Bras-Amoros, 2008)

For all g, we have ng ≥ ng−1. Not true for mf = # of numerical semigroups with Frobenius number f m11 = 51 m12 = 40 m13 = 106

Christopher O’Neill (SDSU) Numerical semigroup: a sales pitch January 17, 2020 11 / 14

SLIDE 55

A couple of long-standing (hard) conjectures

Wilf’s Conjecture

For any S = n1, . . . , nk, we have F(S) + 1 ≤ k(F(S) + 1 − g(S)).

Christopher O’Neill (SDSU) Numerical semigroup: a sales pitch January 17, 2020 12 / 14

SLIDE 56

A couple of long-standing (hard) conjectures

Wilf’s Conjecture

For any S = n1, . . . , nk, we have F(S) + 1 ≤ k(F(S) + 1 − g(S)). Equivalently, 1 k ≤ F(S) + 1 − g(S) F(S) + 1

% of [0, F(S)] in S

Christopher O’Neill (SDSU) Numerical semigroup: a sales pitch January 17, 2020 12 / 14

SLIDE 57

A couple of long-standing (hard) conjectures

Wilf’s Conjecture

For any S = n1, . . . , nk, we have F(S) + 1 ≤ k(F(S) + 1 − g(S)). Equivalently, 1 k ≤ F(S) + 1 − g(S) F(S) + 1

% of [0, F(S)] in S

Christopher O’Neill (SDSU) Numerical semigroup: a sales pitch January 17, 2020 12 / 14

SLIDE 58

A couple of long-standing (hard) conjectures

Wilf’s Conjecture

For any S = n1, . . . , nk, we have F(S) + 1 ≤ k(F(S) + 1 − g(S)). Equivalently, 1 k ≤ F(S) + 1 − g(S) F(S) + 1

% of [0, F(S)] in S

Equality holds when:

Christopher O’Neill (SDSU) Numerical semigroup: a sales pitch January 17, 2020 12 / 14

SLIDE 59

A couple of long-standing (hard) conjectures

Wilf’s Conjecture

For any S = n1, . . . , nk, we have F(S) + 1 ≤ k(F(S) + 1 − g(S)). Equivalently, 1 k ≤ F(S) + 1 − g(S) F(S) + 1

% of [0, F(S)] in S

Equality holds when: S = a, b

Christopher O’Neill (SDSU) Numerical semigroup: a sales pitch January 17, 2020 12 / 14

SLIDE 60

A couple of long-standing (hard) conjectures

Wilf’s Conjecture

For any S = n1, . . . , nk, we have F(S) + 1 ≤ k(F(S) + 1 − g(S)). Equivalently, 1 k ≤ F(S) + 1 − g(S) F(S) + 1

% of [0, F(S)] in S

Equality holds when: S = a, b S = m, m + 1, . . . , 2m − 1

Christopher O’Neill (SDSU) Numerical semigroup: a sales pitch January 17, 2020 12 / 14

SLIDE 61

A couple of long-standing (hard) conjectures

Wilf’s Conjecture

For any S = n1, . . . , nk, we have F(S) + 1 ≤ k(F(S) + 1 − g(S)). Equivalently, 1 k ≤ F(S) + 1 − g(S) F(S) + 1

% of [0, F(S)] in S

Equality holds when: S = a, b S = m, m + 1, . . . , 2m − 1 Proved in many special cases, including g(S) ≤ 60.

Christopher O’Neill (SDSU) Numerical semigroup: a sales pitch January 17, 2020 12 / 14

SLIDE 62

So much more out there!

Christopher O’Neill (SDSU) Numerical semigroup: a sales pitch January 17, 2020 13 / 14

SLIDE 63

So much more out there!

Computation/algorithms

D. Anderson, S. Chapman, N. Kaplan, and D. Torkornoo,

An algorithm to compute ω-primality in a numerical monoid, Semigroup Forum 82 (2011), no. 1, 96–108. [doi]

T. Barron, C. O’Neill, and R. Pelayo,

On dynamic algorithms for factorization invariants in numerical monoids, Mathematics of Computation 86 (2017), 2429–2447. [arXiv:1507.07435]

P. Garc´

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

J. Algebra and its Applications 18 (2019), no. 1, 1950019, 21 pp. [arXiv:1504.02998]

Christopher O’Neill (SDSU) Numerical semigroup: a sales pitch January 17, 2020 13 / 14

SLIDE 64

So much more out there!

Computation/algorithms

D. Anderson, S. Chapman, N. Kaplan, and D. Torkornoo,

An algorithm to compute ω-primality in a numerical monoid, Semigroup Forum 82 (2011), no. 1, 96–108. [doi]

T. Barron, C. O’Neill, and R. Pelayo,

On dynamic algorithms for factorization invariants in numerical monoids, Mathematics of Computation 86 (2017), 2429–2447. [arXiv:1507.07435]

P. Garc´

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

J. Algebra and its Applications 18 (2019), no. 1, 1950019, 21 pp. [arXiv:1504.02998]

Lattice point enumeration and Ehrhart theory (Kunz polyhedra)

J. Autry, A. Ezell, T. Gomes, C. O’Neill, C. Preuss, T. Saluja, and E. Torres-Davila,

Numerical semigroups and polyhedra faces II: locating certain families of semigroups,

preprint. [arXiv:1912.04460]

Christopher O’Neill (SDSU) Numerical semigroup: a sales pitch January 17, 2020 13 / 14

SLIDE 65

So much more out there!

Computation/algorithms

D. Anderson, S. Chapman, N. Kaplan, and D. Torkornoo,

An algorithm to compute ω-primality in a numerical monoid, Semigroup Forum 82 (2011), no. 1, 96–108. [doi]

T. Barron, C. O’Neill, and R. Pelayo,

On dynamic algorithms for factorization invariants in numerical monoids, Mathematics of Computation 86 (2017), 2429–2447. [arXiv:1507.07435]

P. Garc´

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

J. Algebra and its Applications 18 (2019), no. 1, 1950019, 21 pp. [arXiv:1504.02998]

Lattice point enumeration and Ehrhart theory (Kunz polyhedra)

J. Autry, A. Ezell, T. Gomes, C. O’Neill, C. Preuss, T. Saluja, and E. Torres-Davila,

Numerical semigroups and polyhedra faces II: locating certain families of semigroups,

preprint. [arXiv:1912.04460]

Factorization theory

C. Bowles, S. Chapman, N. Kaplan, D. Reiser,

On delta sets of numerical monoids,

J. Algebra Appl. 5 (2006) 1–24. [doi]
J. Amos, S. Chapman, N. Hine, J. Paix˜

ao, Sets of lengths do not characterize numerical monoids, Integers 7 (2007), #A50. [link] Christopher O’Neill (SDSU) Numerical semigroup: a sales pitch January 17, 2020 13 / 14

SLIDE 66

So much more out there!

Computation/algorithms

D. Anderson, S. Chapman, N. Kaplan, and D. Torkornoo,

An algorithm to compute ω-primality in a numerical monoid, Semigroup Forum 82 (2011), no. 1, 96–108. [doi]

T. Barron, C. O’Neill, and R. Pelayo,

On dynamic algorithms for factorization invariants in numerical monoids, Mathematics of Computation 86 (2017), 2429–2447. [arXiv:1507.07435]

P. Garc´

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

J. Algebra and its Applications 18 (2019), no. 1, 1950019, 21 pp. [arXiv:1504.02998]

Lattice point enumeration and Ehrhart theory (Kunz polyhedra)

J. Autry, A. Ezell, T. Gomes, C. O’Neill, C. Preuss, T. Saluja, and E. Torres-Davila,

Numerical semigroups and polyhedra faces II: locating certain families of semigroups,

preprint. [arXiv:1912.04460]

Factorization theory

C. Bowles, S. Chapman, N. Kaplan, D. Reiser,

On delta sets of numerical monoids,

J. Algebra Appl. 5 (2006) 1–24. [doi]
J. Amos, S. Chapman, N. Hine, J. Paix˜

ao, Sets of lengths do not characterize numerical monoids, Integers 7 (2007), #A50. [link]

Enumerative combinatorics

J. Glenn, C. O’Neill, V. Ponomarenko, and B. Sepanski,

Augmented Hilbert series of numerical semigroups, Integers 19 (2019), #A32. [arXiv:1806.11148] Christopher O’Neill (SDSU) Numerical semigroup: a sales pitch January 17, 2020 13 / 14

SLIDE 67

So much more out there!

Computation/algorithms

D. Anderson, S. Chapman, N. Kaplan, and D. Torkornoo,

An algorithm to compute ω-primality in a numerical monoid, Semigroup Forum 82 (2011), no. 1, 96–108. [doi]

T. Barron, C. O’Neill, and R. Pelayo,

On dynamic algorithms for factorization invariants in numerical monoids, Mathematics of Computation 86 (2017), 2429–2447. [arXiv:1507.07435]

P. Garc´

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

J. Algebra and its Applications 18 (2019), no. 1, 1950019, 21 pp. [arXiv:1504.02998]

Lattice point enumeration and Ehrhart theory (Kunz polyhedra)

J. Autry, A. Ezell, T. Gomes, C. O’Neill, C. Preuss, T. Saluja, and E. Torres-Davila,

Numerical semigroups and polyhedra faces II: locating certain families of semigroups,

preprint. [arXiv:1912.04460]

Factorization theory

C. Bowles, S. Chapman, N. Kaplan, D. Reiser,

On delta sets of numerical monoids,

J. Algebra Appl. 5 (2006) 1–24. [doi]
J. Amos, S. Chapman, N. Hine, J. Paix˜

ao, Sets of lengths do not characterize numerical monoids, Integers 7 (2007), #A50. [link]

Enumerative combinatorics

J. Glenn, C. O’Neill, V. Ponomarenko, and B. Sepanski,

Augmented Hilbert series of numerical semigroups, Integers 19 (2019), #A32. [arXiv:1806.11148]

Error-correcting codes (Arf numerical semigroups)

Christopher O’Neill (SDSU) Numerical semigroup: a sales pitch January 17, 2020 13 / 14

SLIDE 68

So much more out there!

Computation/algorithms

D. Anderson, S. Chapman, N. Kaplan, and D. Torkornoo,

An algorithm to compute ω-primality in a numerical monoid, Semigroup Forum 82 (2011), no. 1, 96–108. [doi]

T. Barron, C. O’Neill, and R. Pelayo,

On dynamic algorithms for factorization invariants in numerical monoids, Mathematics of Computation 86 (2017), 2429–2447. [arXiv:1507.07435]

P. Garc´

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

J. Algebra and its Applications 18 (2019), no. 1, 1950019, 21 pp. [arXiv:1504.02998]

Lattice point enumeration and Ehrhart theory (Kunz polyhedra)

J. Autry, A. Ezell, T. Gomes, C. O’Neill, C. Preuss, T. Saluja, and E. Torres-Davila,

Numerical semigroups and polyhedra faces II: locating certain families of semigroups,

preprint. [arXiv:1912.04460]

Factorization theory

C. Bowles, S. Chapman, N. Kaplan, D. Reiser,

On delta sets of numerical monoids,

J. Algebra Appl. 5 (2006) 1–24. [doi]
J. Amos, S. Chapman, N. Hine, J. Paix˜

ao, Sets of lengths do not characterize numerical monoids, Integers 7 (2007), #A50. [link]

Enumerative combinatorics

J. Glenn, C. O’Neill, V. Ponomarenko, and B. Sepanski,

Augmented Hilbert series of numerical semigroups, Integers 19 (2019), #A32. [arXiv:1806.11148]

Error-correcting codes (Arf numerical semigroups) Music theory

Christopher O’Neill (SDSU) Numerical semigroup: a sales pitch January 17, 2020 13 / 14

SLIDE 69

Diving in headfirst

How can a talented undergraduate I know get started?

Christopher O’Neill (SDSU) Numerical semigroup: a sales pitch January 17, 2020 14 / 14

SLIDE 70

Diving in headfirst

How can a talented undergraduate I know get started? Survey papers (several include open problems)

C. O’Neill and R. Pelayo,

How do you measure primality?, American Mathematical Monthly 122 (2015), no. 2, 121–137. [arXiv:1405.1714]

C. O’Neill and R. Pelayo,

Factorization invariants in numerical monoids, Contemporary Mathematics 685 (2017), 231–249. [arXiv:1508.00128]

S. Chapman and C. O’Neill,

Factoring in the Chicken McNugget monoid, Mathematics Magazine 91 (2018), no. 5, 323–336. [arXiv:1709.01606]

S. Chapman, R. Garcia, and C. O’Neill,

Beyond coins, stamps, and Chicken McNuggets: an invitation to numerical semigroups, to appear, FURM Volume 3 (Springer). [arXiv:1902.05848] Christopher O’Neill (SDSU) Numerical semigroup: a sales pitch January 17, 2020 14 / 14

SLIDE 71

Diving in headfirst. . . and a shameless plug

How can a talented undergraduate I know get started? Survey papers (several include open problems)

C. O’Neill and R. Pelayo,

How do you measure primality?, American Mathematical Monthly 122 (2015), no. 2, 121–137. [arXiv:1405.1714]

C. O’Neill and R. Pelayo,

Factorization invariants in numerical monoids, Contemporary Mathematics 685 (2017), 231–249. [arXiv:1508.00128]

S. Chapman and C. O’Neill,

Factoring in the Chicken McNugget monoid, Mathematics Magazine 91 (2018), no. 5, 323–336. [arXiv:1709.01606]

S. Chapman, R. Garcia, and C. O’Neill,

Beyond coins, stamps, and Chicken McNuggets: an invitation to numerical semigroups, to appear, FURM Volume 3 (Springer). [arXiv:1902.05848]

Apply to the San Diego State University summer REU!!!! Faculty expert: Maria Bras-Amoros

(Universitat Rovira i Virgili, in Catalonia)

Apply: http://www.sci.sdsu.edu/math-reu/ Deadline: March 1st, 2020

Christopher O’Neill (SDSU) Numerical semigroup: a sales pitch January 17, 2020 14 / 14