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Enumerating numerical semigroups using polyhedral geometry - - PowerPoint PPT Presentation

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Enumerating numerical semigroups using polyhedral geometry Christopher ONeill San Diego State University cdoneill@sdsu.edu Joint with Winfried Bruns, Pedro Garc a S anchez, and Dane Wilburne May 4, 2019 Christopher ONeill (SDSU)

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Enumerating numerical semigroups using polyhedral geometry

Christopher O’Neill

San Diego State University cdoneill@sdsu.edu Joint with Winfried Bruns, Pedro Garc´ ıa S´ anchez, and Dane Wilburne

May 4, 2019

Christopher O’Neill (SDSU) Enumerating numerical semigroups May 4, 2019 1 / 18

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Numerical semigroups

Definition

A numerical semigroup S ⊂ Z≥0: closed under addition, |Z≥0 \ S| < ∞.

Christopher O’Neill (SDSU) Enumerating numerical semigroups May 4, 2019 2 / 18

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Numerical semigroups

Definition

A numerical semigroup S ⊂ Z≥0: closed under addition, |Z≥0 \ S| < ∞. Example: McN = 6, 9, 20 =

  • 0, 6, 9, 12, 15, 18, 20, 21, 24, . . .

. . . , 36, 38, 39, 40, 41, 42, 44 →

  • Christopher O’Neill (SDSU)

Enumerating numerical semigroups May 4, 2019 2 / 18

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Numerical semigroups

Definition

A numerical semigroup S ⊂ Z≥0: closed under addition, |Z≥0 \ S| < ∞. Example: “McNugget Semigroup” McN = 6, 9, 20 =

  • 0, 6, 9, 12, 15, 18, 20, 21, 24, . . .

. . . , 36, 38, 39, 40, 41, 42, 44 →

  • Christopher O’Neill (SDSU)

Enumerating numerical semigroups May 4, 2019 2 / 18

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Numerical semigroups

Definition

A numerical semigroup S ⊂ Z≥0: closed under addition, |Z≥0 \ S| < ∞. Example: “McNugget Semigroup” McN = 6, 9, 20 =

  • 0, 6, 9, 12, 15, 18, 20, 21, 24, . . .

. . . , 36, 38, 39, 40, 41, 42, 44 →

  • Example: S = 6, 9, 18, 20, 32

Christopher O’Neill (SDSU) Enumerating numerical semigroups May 4, 2019 2 / 18

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Numerical semigroups

Definition

A numerical semigroup S ⊂ Z≥0: closed under addition, |Z≥0 \ S| < ∞. Example: “McNugget Semigroup” McN = 6, 9, 20 =

  • 0, 6, 9, 12, 15, 18, 20, 21, 24, . . .

. . . , 36, 38, 39, 40, 41, 42, 44 →

  • Example: S = 6, 9, 18, 20, 32

Christopher O’Neill (SDSU) Enumerating numerical semigroups May 4, 2019 2 / 18

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Numerical semigroups

Definition

A numerical semigroup S ⊂ Z≥0: closed under addition, |Z≥0 \ S| < ∞. Example: “McNugget Semigroup” McN = 6, 9, 20 =

  • 0, 6, 9, 12, 15, 18, 20, 21, 24, . . .

. . . , 36, 38, 39, 40, 41, 42, 44 →

  • Example: S = 6, 9, 18, 20, 32 = McN

Christopher O’Neill (SDSU) Enumerating numerical semigroups May 4, 2019 2 / 18

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Numerical semigroups

Definition

A numerical semigroup S ⊂ Z≥0: closed under addition, |Z≥0 \ S| < ∞. Example: “McNugget Semigroup” McN = 6, 9, 20 =

  • 0, 6, 9, 12, 15, 18, 20, 21, 24, . . .

. . . , 36, 38, 39, 40, 41, 42, 44 →

  • Example: S = 6, 9, 18, 20, 32 = McN

Fact

Every numerical semigroup has a unique minimal generating set.

Christopher O’Neill (SDSU) Enumerating numerical semigroups May 4, 2019 2 / 18

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Numerical semigroups

Definition

A numerical semigroup S ⊂ Z≥0: closed under addition, |Z≥0 \ S| < ∞. Example: “McNugget Semigroup” McN = 6, 9, 20 =

  • 0, 6, 9, 12, 15, 18, 20, 21, 24, . . .

. . . , 36, 38, 39, 40, 41, 42, 44 →

  • Example: S = 6, 9, 18, 20, 32 = McN

Fact

Every numerical semigroup has a unique minimal generating set. Embedding dimension: e(S) = # minimal generators

Christopher O’Neill (SDSU) Enumerating numerical semigroups May 4, 2019 2 / 18

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Numerical semigroups

Definition

A numerical semigroup S ⊂ Z≥0: closed under addition, |Z≥0 \ S| < ∞. Example: “McNugget Semigroup” McN = 6, 9, 20 =

  • 0, 6, 9, 12, 15, 18, 20, 21, 24, . . .

. . . , 36, 38, 39, 40, 41, 42, 44 →

  • Example: S = 6, 9, 18, 20, 32 = McN

Fact

Every numerical semigroup has a unique minimal generating set. Embedding dimension: e(S) = # minimal generators Multiplicity: m(S) = smallest nonzero element

Christopher O’Neill (SDSU) Enumerating numerical semigroups May 4, 2019 2 / 18

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Frobenius number

Fix a numerical semigroup S = n1, . . . , nk.

Christopher O’Neill (SDSU) Enumerating numerical semigroups May 4, 2019 3 / 18

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Frobenius number

Fix a numerical semigroup S = n1, . . . , nk.

Definition

F(S) = max(N \ S) is the Frobenius number of S.

Christopher O’Neill (SDSU) Enumerating numerical semigroups May 4, 2019 3 / 18

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Frobenius number

Fix a numerical semigroup S = n1, . . . , nk.

Definition

F(S) = max(N \ S) is the Frobenius number of S.

Example

If S = 6, 9, 20, then F(S) = 43 since N \ S = {1, 2, 3, 4, 5, 7, 8, 10, 11, 13, . . . , 31, 34, 37, 43}.

Christopher O’Neill (SDSU) Enumerating numerical semigroups May 4, 2019 3 / 18

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Frobenius number

Fix a numerical semigroup S = n1, . . . , nk.

Definition

F(S) = max(N \ S) is the Frobenius number of S.

Example

If S = 6, 9, 20, then F(S) = 43 since N \ S = {1, 2, 3, 4, 5, 7, 8, 10, 11, 13, . . . , 31, 34, 37, 43}. Computing the Frobenius number for general S is hard.

Christopher O’Neill (SDSU) Enumerating numerical semigroups May 4, 2019 3 / 18

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Frobenius number

Fix a numerical semigroup S = n1, . . . , nk.

Definition

F(S) = max(N \ S) is the Frobenius number of S.

Example

If S = 6, 9, 20, then F(S) = 43 since N \ S = {1, 2, 3, 4, 5, 7, 8, 10, 11, 13, . . . , 31, 34, 37, 43}. Computing the Frobenius number for general S is hard. If S = n1, n2, then F(S) = n1n2 − (n1 + n2).

Christopher O’Neill (SDSU) Enumerating numerical semigroups May 4, 2019 3 / 18

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Frobenius number

Fix a numerical semigroup S = n1, . . . , nk.

Definition

F(S) = max(N \ S) is the Frobenius number of S.

Example

If S = 6, 9, 20, then F(S) = 43 since N \ S = {1, 2, 3, 4, 5, 7, 8, 10, 11, 13, . . . , 31, 34, 37, 43}. Computing the Frobenius number for general S is hard. If S = n1, n2, then F(S) = n1n2 − (n1 + n2). If S = n1, n2, n3, then there is a fast algorithm for F(S).

Christopher O’Neill (SDSU) Enumerating numerical semigroups May 4, 2019 3 / 18

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Frobenius number

Fix a numerical semigroup S = n1, . . . , nk.

Definition

F(S) = max(N \ S) is the Frobenius number of S.

Example

If S = 6, 9, 20, then F(S) = 43 since N \ S = {1, 2, 3, 4, 5, 7, 8, 10, 11, 13, . . . , 31, 34, 37, 43}. Computing the Frobenius number for general S is hard. If S = n1, n2, then F(S) = n1n2 − (n1 + n2). If S = n1, n2, n3, then there is a fast algorithm for F(S). Formulas in a few other special cases.

Christopher O’Neill (SDSU) Enumerating numerical semigroups May 4, 2019 3 / 18

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The Ap´ ery set

Fix a numerical semigroup S with m(S) = m.

Christopher O’Neill (SDSU) Enumerating numerical semigroups May 4, 2019 4 / 18

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The Ap´ ery set

Fix a numerical semigroup S with m(S) = m.

Definition

The Ap´ ery set of S is Ap(S) = {a ∈ S : a − m / ∈ S}

Christopher O’Neill (SDSU) Enumerating numerical semigroups May 4, 2019 4 / 18

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The Ap´ ery set

Fix a numerical semigroup S with m(S) = m.

Definition

The Ap´ ery set of S is Ap(S) = {a ∈ S : a − m / ∈ S} If S = 6, 9, 20, then Ap(S) = {0, 49, 20, 9, 40, 29}.

Christopher O’Neill (SDSU) Enumerating numerical semigroups May 4, 2019 4 / 18

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The Ap´ ery set

Fix a numerical semigroup S with m(S) = m.

Definition

The Ap´ ery set of S is Ap(S) = {a ∈ S : a − m / ∈ S} If S = 6, 9, 20, then Ap(S) = {0, 49, 20, 9, 40, 29}. For 2 mod 6: {2, 8, 14, 20, 26, 32, . . .} ∩ S = {20, 26, 32, . . .} For 3 mod 6: {3, 9, 15, 21, . . .} ∩ S = {9, 15, 21, . . .} For 4 mod 6: {4, 10, 16, 22, . . .} ∩ S = {40, 46, 52, . . .}

Christopher O’Neill (SDSU) Enumerating numerical semigroups May 4, 2019 4 / 18

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The Ap´ ery set

Fix a numerical semigroup S with m(S) = m.

Definition

The Ap´ ery set of S is Ap(S) = {a ∈ S : a − m / ∈ S} If S = 6, 9, 20, then Ap(S) = {0, 49, 20, 9, 40, 29}. For 2 mod 6: {2, 8, 14, 20, 26, 32, . . .} ∩ S = {20, 26, 32, . . .} For 3 mod 6: {3, 9, 15, 21, . . .} ∩ S = {9, 15, 21, . . .} For 4 mod 6: {4, 10, 16, 22, . . .} ∩ S = {40, 46, 52, . . .}

Christopher O’Neill (SDSU) Enumerating numerical semigroups May 4, 2019 4 / 18

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The Ap´ ery set

Fix a numerical semigroup S with m(S) = m.

Definition

The Ap´ ery set of S is Ap(S) = {a ∈ S : a − m / ∈ S} If S = 6, 9, 20, then Ap(S) = {0, 49, 20, 9, 40, 29}. For 2 mod 6: {2, 8, 14, 20, 26, 32, . . .} ∩ S = {20, 26, 32, . . .} For 3 mod 6: {3, 9, 15, 21, . . .} ∩ S = {9, 15, 21, . . .} For 4 mod 6: {4, 10, 16, 22, . . .} ∩ S = {40, 46, 52, . . .}

Christopher O’Neill (SDSU) Enumerating numerical semigroups May 4, 2019 4 / 18

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The Ap´ ery set

Fix a numerical semigroup S with m(S) = m.

Definition

The Ap´ ery set of S is Ap(S) = {a ∈ S : a − m / ∈ S} If S = 6, 9, 20, then Ap(S) = {0, 49, 20, 9, 40, 29}. For 2 mod 6: {2, 8, 14, 20, 26, 32, . . .} ∩ S = {20, 26, 32, . . .} For 3 mod 6: {3, 9, 15, 21, . . .} ∩ S = {9, 15, 21, . . .} For 4 mod 6: {4, 10, 16, 22, . . .} ∩ S = {40, 46, 52, . . .} Observations:

Christopher O’Neill (SDSU) Enumerating numerical semigroups May 4, 2019 4 / 18

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The Ap´ ery set

Fix a numerical semigroup S with m(S) = m.

Definition

The Ap´ ery set of S is Ap(S) = {a ∈ S : a − m / ∈ S} If S = 6, 9, 20, then Ap(S) = {0, 49, 20, 9, 40, 29}. For 2 mod 6: {2, 8, 14, 20, 26, 32, . . .} ∩ S = {20, 26, 32, . . .} For 3 mod 6: {3, 9, 15, 21, . . .} ∩ S = {9, 15, 21, . . .} For 4 mod 6: {4, 10, 16, 22, . . .} ∩ S = {40, 46, 52, . . .} Observations: The elements of Ap(S) are distinct modulo m

Christopher O’Neill (SDSU) Enumerating numerical semigroups May 4, 2019 4 / 18

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The Ap´ ery set

Fix a numerical semigroup S with m(S) = m.

Definition

The Ap´ ery set of S is Ap(S) = {a ∈ S : a − m / ∈ S} If S = 6, 9, 20, then Ap(S) = {0, 49, 20, 9, 40, 29}. For 2 mod 6: {2, 8, 14, 20, 26, 32, . . .} ∩ S = {20, 26, 32, . . .} For 3 mod 6: {3, 9, 15, 21, . . .} ∩ S = {9, 15, 21, . . .} For 4 mod 6: {4, 10, 16, 22, . . .} ∩ S = {40, 46, 52, . . .} Observations: The elements of Ap(S) are distinct modulo m | Ap(S)| = m

Christopher O’Neill (SDSU) Enumerating numerical semigroups May 4, 2019 4 / 18

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The Ap´ ery set

Fix a numerical semigroup S with m(S) = m.

Definition

The Ap´ ery set of S is Ap(S) = {a ∈ S : a − m / ∈ S}

Christopher O’Neill (SDSU) Enumerating numerical semigroups May 4, 2019 5 / 18

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The Ap´ ery set

Fix a numerical semigroup S with m(S) = m.

Definition

The Ap´ ery set of S is Ap(S) = {a ∈ S : a − m / ∈ S} Many things can be easily recovered from the Ap´ ery set.

Christopher O’Neill (SDSU) Enumerating numerical semigroups May 4, 2019 5 / 18

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The Ap´ ery set

Fix a numerical semigroup S with m(S) = m.

Definition

The Ap´ ery set of S is Ap(S) = {a ∈ S : a − m / ∈ S} Many things can be easily recovered from the Ap´ ery set. Fast membership test: n ∈ S if n ≥ a for a ∈ Ap(S) with a ≡ n mod m

Christopher O’Neill (SDSU) Enumerating numerical semigroups May 4, 2019 5 / 18

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The Ap´ ery set

Fix a numerical semigroup S with m(S) = m.

Definition

The Ap´ ery set of S is Ap(S) = {a ∈ S : a − m / ∈ S} Many things can be easily recovered from the Ap´ ery set. Fast membership test: n ∈ S if n ≥ a for a ∈ Ap(S) with a ≡ n mod m Frobenius number: F(S) = max(Ap(S)) − m

Christopher O’Neill (SDSU) Enumerating numerical semigroups May 4, 2019 5 / 18

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The Ap´ ery set

Fix a numerical semigroup S with m(S) = m.

Definition

The Ap´ ery set of S is Ap(S) = {a ∈ S : a − m / ∈ S} Many things can be easily recovered from the Ap´ ery set. Fast membership test: n ∈ S if n ≥ a for a ∈ Ap(S) with a ≡ n mod m Frobenius number: F(S) = max(Ap(S)) − m Number of gaps (the genus): g(S) = |N \ S| =

  • a∈Ap(S)

a

m

  • Christopher O’Neill (SDSU)

Enumerating numerical semigroups May 4, 2019 5 / 18

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The Ap´ ery set

Fix a numerical semigroup S with m(S) = m.

Definition

The Ap´ ery set of S is Ap(S) = {a ∈ S : a − m / ∈ S} Many things can be easily recovered from the Ap´ ery set. Fast membership test: n ∈ S if n ≥ a for a ∈ Ap(S) with a ≡ n mod m Frobenius number: F(S) = max(Ap(S)) − m Number of gaps (the genus): g(S) = |N \ S| =

  • a∈Ap(S)

a

m

  • The Ap´

ery set is a “one stop shop” for computation.

Christopher O’Neill (SDSU) Enumerating numerical semigroups May 4, 2019 5 / 18

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The Ap´ ery poset

Fix a numerical semigroup S with m(S) = m.

Definition

The Ap´ ery set of S is Ap(S) = {a ∈ S : a − m / ∈ S}

Christopher O’Neill (SDSU) Enumerating numerical semigroups May 4, 2019 6 / 18

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The Ap´ ery poset

Fix a numerical semigroup S with m(S) = m.

Definition

The Ap´ ery set of S is Ap(S) = {a ∈ S : a − m / ∈ S} If S = 6, 9, 20, then Ap(S) = {0, 49, 20, 9, 40, 29}.

Christopher O’Neill (SDSU) Enumerating numerical semigroups May 4, 2019 6 / 18

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The Ap´ ery poset

Fix a numerical semigroup S with m(S) = m.

Definition

The Ap´ ery set of S is Ap(S) = {a ∈ S : a − m / ∈ S} If S = 6, 9, 20, then Ap(S) = {0, 49, 20, 9, 40, 29}. The Ap´ ery poset of S: define a b whenever b − a ∈ S.

Christopher O’Neill (SDSU) Enumerating numerical semigroups May 4, 2019 6 / 18

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The Ap´ ery poset

Fix a numerical semigroup S with m(S) = m.

Definition

The Ap´ ery set of S is Ap(S) = {a ∈ S : a − m / ∈ S} If S = 6, 9, 20, then Ap(S) = {0, 49, 20, 9, 40, 29}. The Ap´ ery poset of S: define a b whenever b − a ∈ S.

Christopher O’Neill (SDSU) Enumerating numerical semigroups May 4, 2019 6 / 18

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The Ap´ ery poset

Fix a numerical semigroup S with m(S) = m.

Definition

The Ap´ ery set of S is Ap(S) = {a ∈ S : a − m / ∈ S} If S = 6, 9, 20, then Ap(S) = {0, 49, 20, 9, 40, 29}. The Ap´ ery poset of S: define a b whenever b − a ∈ S. e(S) = # min elements + 1

Christopher O’Neill (SDSU) Enumerating numerical semigroups May 4, 2019 6 / 18

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A tantalizing conjecture

Wilf’s Conjecture

For any numerical semigroup S, F(S) + 1 ≤ e(S)(F(S) + 1 − g(S)).

Christopher O’Neill (SDSU) Enumerating numerical semigroups May 4, 2019 7 / 18

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A tantalizing conjecture

Wilf’s Conjecture

For any numerical semigroup S, F(S) + 1 ≤ e(S)(F(S) + 1 − g(S)). Equivalently, 1 e(S) ≤ F(S) + 1 − g(S) F(S) + 1

  • % of [0, F(S)] in S

Christopher O’Neill (SDSU) Enumerating numerical semigroups May 4, 2019 7 / 18

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A tantalizing conjecture

Wilf’s Conjecture

For any numerical semigroup S, F(S) + 1 ≤ e(S)(F(S) + 1 − g(S)). Equivalently, 1 e(S) ≤ F(S) + 1 − g(S) F(S) + 1

  • % of [0, F(S)] in S

Christopher O’Neill (SDSU) Enumerating numerical semigroups May 4, 2019 7 / 18

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A tantalizing conjecture

Wilf’s Conjecture

For any numerical semigroup S, F(S) + 1 ≤ e(S)(F(S) + 1 − g(S)). Equivalently, 1 e(S) ≤ F(S) + 1 − g(S) F(S) + 1

  • % of [0, F(S)] in S

Equality holds when:

Christopher O’Neill (SDSU) Enumerating numerical semigroups May 4, 2019 7 / 18

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A tantalizing conjecture

Wilf’s Conjecture

For any numerical semigroup S, F(S) + 1 ≤ e(S)(F(S) + 1 − g(S)). Equivalently, 1 e(S) ≤ F(S) + 1 − g(S) F(S) + 1

  • % of [0, F(S)] in S

Equality holds when: S = a, b

Christopher O’Neill (SDSU) Enumerating numerical semigroups May 4, 2019 7 / 18

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A tantalizing conjecture

Wilf’s Conjecture

For any numerical semigroup S, F(S) + 1 ≤ e(S)(F(S) + 1 − g(S)). Equivalently, 1 e(S) ≤ 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 (maximal embedding dimension)

Christopher O’Neill (SDSU) Enumerating numerical semigroups May 4, 2019 7 / 18

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A tantalizing conjecture

Wilf’s Conjecture

For any numerical semigroup S, F(S) + 1 ≤ e(S)(F(S) + 1 − g(S)). Equivalently, 1 e(S) ≤ 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 (maximal embedding dimension) Proved in many special cases, including g(S) ≤ 60.

Christopher O’Neill (SDSU) Enumerating numerical semigroups May 4, 2019 7 / 18

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Polyhedral geometry enters the picture

Fix a numerical semigroup S with m(S) = m, and write Ap(S) = {0, a1, . . . , am−1} with ai = mxi + i.

Christopher O’Neill (SDSU) Enumerating numerical semigroups May 4, 2019 8 / 18

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Polyhedral geometry enters the picture

Fix a numerical semigroup S with m(S) = m, and write Ap(S) = {0, a1, . . . , am−1} with ai = mxi + i. The Kunz coordinates of S: (x1, . . . , xm−1) ∈ Z≥1.

Christopher O’Neill (SDSU) Enumerating numerical semigroups May 4, 2019 8 / 18

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Polyhedral geometry enters the picture

Fix a numerical semigroup S with m(S) = m, and write Ap(S) = {0, a1, . . . , am−1} with ai = mxi + i. The Kunz coordinates of S: (x1, . . . , xm−1) ∈ Z≥1. Key observation: ai + aj ≥ ai+j

Christopher O’Neill (SDSU) Enumerating numerical semigroups May 4, 2019 8 / 18

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SLIDE 48

Polyhedral geometry enters the picture

Fix a numerical semigroup S with m(S) = m, and write Ap(S) = {0, a1, . . . , am−1} with ai = mxi + i. The Kunz coordinates of S: (x1, . . . , xm−1) ∈ Z≥1. Key observation: ai + aj ≥ ai+j

Theorem (Kunz)

A point (x1, . . . , xm−1) ∈ Zm−1 is the Kunz coordinates of a numerical semigroup if and only if for 1 ≤ i, j ≤ m − 1, xi ≥ 1 xi + xj ≥ xi+j for i + j < m 1 + xi + xj ≥ xi+j−m for i + j > m

Christopher O’Neill (SDSU) Enumerating numerical semigroups May 4, 2019 8 / 18

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SLIDE 49

Polyhedral geometry enters the picture

Fix a numerical semigroup S with m(S) = m, and write Ap(S) = {0, a1, . . . , am−1} with ai = mxi + i. The Kunz coordinates of S: (x1, . . . , xm−1) ∈ Z≥1. Key observation: ai + aj ≥ ai+j

Theorem (Kunz)

A point (x1, . . . , xm−1) ∈ Zm−1 is the Kunz coordinates of a numerical semigroup if and only if for 1 ≤ i, j ≤ m − 1, xi ≥ 1 xi + xj ≥ xi+j for i + j < m 1 + xi + xj ≥ xi+j−m for i + j > m Numerical semigroups ← → integer points in rational polyhedra!

Christopher O’Neill (SDSU) Enumerating numerical semigroups May 4, 2019 8 / 18

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Kunz polyhedra

Kunz polyhedron P3 ⊂ R2

Christopher O’Neill (SDSU) Enumerating numerical semigroups May 4, 2019 9 / 18

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Kunz polyhedra

Kunz polyhedron P3 ⊂ R2 Example: (4, 3) ∈ P3 Ap(S) = {0, 3·4+1, 3·3+2} ⇒ S = 3, 11, 13

Christopher O’Neill (SDSU) Enumerating numerical semigroups May 4, 2019 9 / 18

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Kunz polyhedra

Kunz polyhedron P3 ⊂ R2 Example: (4, 3) ∈ P3 Ap(S) = {0, 3·4+1, 3·3+2} ⇒ S = 3, 11, 13 Observations:

Christopher O’Neill (SDSU) Enumerating numerical semigroups May 4, 2019 9 / 18

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Kunz polyhedra

Kunz polyhedron P3 ⊂ R2 Example: (4, 3) ∈ P3 Ap(S) = {0, 3·4+1, 3·3+2} ⇒ S = 3, 11, 13 Observations: g(S) = x1 + · · · + xm−1

Christopher O’Neill (SDSU) Enumerating numerical semigroups May 4, 2019 9 / 18

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Kunz polyhedra

Kunz polyhedron P3 ⊂ R2 Example: (4, 3) ∈ P3 Ap(S) = {0, 3·4+1, 3·3+2} ⇒ S = 3, 11, 13 Observations: g(S) = x1 + · · · + xm−1 F(S) = max{mxi +i}−m

Christopher O’Neill (SDSU) Enumerating numerical semigroups May 4, 2019 9 / 18

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Kunz polyhedra

Kunz polyhedron P3 ⊂ R2 Example: (4, 3) ∈ P3 Ap(S) = {0, 3·4+1, 3·3+2} ⇒ S = 3, 11, 13 Observations: g(S) = x1 + · · · + xm−1 F(S) = max{mxi +i}−m

Christopher O’Neill (SDSU) Enumerating numerical semigroups May 4, 2019 9 / 18

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SLIDE 56

Kunz polyhedra

Kunz polyhedron P3 ⊂ R2 Example: (4, 3) ∈ P3 Ap(S) = {0, 3·4+1, 3·3+2} ⇒ S = 3, 11, 13 Observations: g(S) = x1 + · · · + xm−1 F(S) = max{mxi +i}−m S max embedding dim when S ∈ Int(Pm)

Christopher O’Neill (SDSU) Enumerating numerical semigroups May 4, 2019 9 / 18

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SLIDE 57

Kunz polyhedra

Kunz polyhedron P3 ⊂ R2 Example: (4, 3) ∈ P3 Ap(S) = {0, 3·4+1, 3·3+2} ⇒ S = 3, 11, 13 Observations: g(S) = x1 + · · · + xm−1 F(S) = max{mxi +i}−m S max embedding dim when S ∈ Int(Pm)

Christopher O’Neill (SDSU) Enumerating numerical semigroups May 4, 2019 9 / 18

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SLIDE 58

Kunz polyhedra

Kunz Polyhedron P4 ⊂ R3 (boundary only)

Christopher O’Neill (SDSU) Enumerating numerical semigroups May 4, 2019 10 / 18

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SLIDE 59

Faces of the Kunz polyhedron

Question

When are 2 numerical semigroups in the relative interior of the same face?

Christopher O’Neill (SDSU) Enumerating numerical semigroups May 4, 2019 11 / 18

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SLIDE 60

Faces of the Kunz polyhedron

Question

When are 2 numerical semigroups in the relative interior of the same face? S = 6, 9, 20 Ap(S) = {0, 49, 20, 9, 40, 29} S = 6, 26, 27 Ap(S) = {0, 79, 26, 27, 52, 53}

Christopher O’Neill (SDSU) Enumerating numerical semigroups May 4, 2019 11 / 18

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SLIDE 61

Faces of the Kunz polyhedron

Question

When are 2 numerical semigroups in the relative interior of the same face? S = 6, 9, 20 Ap(S) = {0, 49, 20, 9, 40, 29} S = 6, 26, 27 Ap(S) = {0, 79, 26, 27, 52, 53}

Christopher O’Neill (SDSU) Enumerating numerical semigroups May 4, 2019 11 / 18

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SLIDE 62

Faces of the Kunz polyhedron

Question

When are 2 numerical semigroups in the relative interior of the same face? S = 6, 9, 20 Ap(S) = {0, 49, 20, 9, 40, 29}

1 2 3 4 5

S = 6, 26, 27 Ap(S) = {0, 79, 26, 27, 52, 53}

1 2 3 4 5

Christopher O’Neill (SDSU) Enumerating numerical semigroups May 4, 2019 11 / 18

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SLIDE 63

Faces of the Kunz polyhedron

Question

When are 2 numerical semigroups in the relative interior of the same face? S = 6, 9, 20 Ap(S) = {0, 49, 20, 9, 40, 29}

1 2 3 4 5

S = 6, 26, 27 Ap(S) = {0, 79, 26, 27, 52, 53}

1 2 3 4 5

The Kunz poset of S: use ground set Zm \ {0} instead of Ap(S) \ {0}.

Christopher O’Neill (SDSU) Enumerating numerical semigroups May 4, 2019 11 / 18

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SLIDE 64

Faces of the Kunz polyhedron

Question

When are 2 numerical semigroups in the relative interior of the same face? S = 6, 9, 20 Ap(S) = {0, 49, 20, 9, 40, 29}

1 2 3 4 5

S = 6, 26, 27 Ap(S) = {0, 79, 26, 27, 52, 53}

1 2 3 4 5

The Kunz poset of S: use ground set Zm \ {0} instead of Ap(S) \ {0}.

Theorem (Bruns, Garc´ ıa-S´ anchez, O., Wilburne)

Two numerical semigroups lie in the relative interior of the same face

  • f Pm if and only if their Kunz posets are identical.

Christopher O’Neill (SDSU) Enumerating numerical semigroups May 4, 2019 11 / 18

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SLIDE 65

Faces of the Kunz polyhedron

Question

When are 2 numerical semigroups in the relative interior of the same face? S = 6, 9, 20 Ap(S) = {0, 49, 20, 9, 40, 29}

1 2 3 4 5

The Kunz poset of S: use ground set Zm \ {0} instead of Ap(S) \ {0}.

Theorem (Bruns, Garc´ ıa-S´ anchez, O., Wilburne)

If two numerical semigroups lie in the relative interior of the same face

  • f Pm, then their Kunz posets are identical.

Christopher O’Neill (SDSU) Enumerating numerical semigroups May 4, 2019 12 / 18

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SLIDE 66

Faces of the Kunz polyhedron

Question

When are 2 numerical semigroups in the relative interior of the same face? S = 6, 9, 20 Ap(S) = {0, 49, 20, 9, 40, 29}

1 2 3 4 5

Defining facet equations: 2x2 = x4 x2 + x3 = x5 x2 + x5 = x1 − 1 x3 + x4 = x1 − 1 The Kunz poset of S: use ground set Zm \ {0} instead of Ap(S) \ {0}.

Theorem (Bruns, Garc´ ıa-S´ anchez, O., Wilburne)

If two numerical semigroups lie in the relative interior of the same face

  • f Pm, then their Kunz posets are identical.

Christopher O’Neill (SDSU) Enumerating numerical semigroups May 4, 2019 12 / 18

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SLIDE 67

Faces of the Kunz polyhedron

Question

When are 2 numerical semigroups in the relative interior of the same face? S = 6, 9, 20 Ap(S) = {0, 49, 20, 9, 40, 29}

1 2 3 4 5

Defining facet equations: 2x2 = x4 x2 + x3 = x5 x2 + x5 = x1 − 1 x3 + x4 = x1 − 1 2 4 2 5 3 5 2 1 5 1 3 1 4 1 The Kunz poset of S: use ground set Zm \ {0} instead of Ap(S) \ {0}.

Theorem (Bruns, Garc´ ıa-S´ anchez, O., Wilburne)

If two numerical semigroups lie in the relative interior of the same face

  • f Pm, then their Kunz posets are identical.

Christopher O’Neill (SDSU) Enumerating numerical semigroups May 4, 2019 12 / 18

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SLIDE 68

Faces of the Kunz polyhedron

Question

When are 2 numerical semigroups in the relative interior of the same face?

Christopher O’Neill (SDSU) Enumerating numerical semigroups May 4, 2019 13 / 18

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SLIDE 69

Faces of the Kunz polyhedron

Question

When are 2 numerical semigroups in the relative interior of the same face?

Christopher O’Neill (SDSU) Enumerating numerical semigroups May 4, 2019 13 / 18

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SLIDE 70

Faces of the Kunz polyhedron

Question

When are 2 numerical semigroups in the relative interior of the same face? S = 3, 7 S = 3, 8 S = 3, 5, 7

Christopher O’Neill (SDSU) Enumerating numerical semigroups May 4, 2019 13 / 18

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SLIDE 71

Faces of the Kunz polyhedron

Question

When are 2 numerical semigroups in the relative interior of the same face? S = 3, 7 S = 3, 8 S = 3, 5, 7

Christopher O’Neill (SDSU) Enumerating numerical semigroups May 4, 2019 13 / 18

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SLIDE 72

Faces of the Kunz polyhedron

Question

When are 2 numerical semigroups in the relative interior of the same face? S = 3, 7 S = 3, 8 S = 3, 5, 7

Christopher O’Neill (SDSU) Enumerating numerical semigroups May 4, 2019 13 / 18

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SLIDE 73

Faces of the Kunz polyhedron

Question

When are 2 numerical semigroups in the relative interior of the same face? S = 3, 7 S = 3, 8 S = 3, 5, 7

Christopher O’Neill (SDSU) Enumerating numerical semigroups May 4, 2019 13 / 18

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SLIDE 74

Faces of the Kunz polyhedron

Question

When are 2 numerical semigroups in the relative interior of the same face? S = 3, 7 S = 3, 8 S = 3, 5, 7

Christopher O’Neill (SDSU) Enumerating numerical semigroups May 4, 2019 13 / 18

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SLIDE 75

Faces of the Kunz polyhedron

Question

When are 2 numerical semigroups in the relative interior of the same face?

Christopher O’Neill (SDSU) Enumerating numerical semigroups May 4, 2019 14 / 18

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SLIDE 76

Faces of the Kunz polyhedron

Question

When are 2 numerical semigroups in the relative interior of the same face?

Christopher O’Neill (SDSU) Enumerating numerical semigroups May 4, 2019 14 / 18

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SLIDE 77

Faces of the Kunz polyhedron

Question

When are 2 numerical semigroups in the relative interior of the same face?

Christopher O’Neill (SDSU) Enumerating numerical semigroups May 4, 2019 14 / 18

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SLIDE 78

Faces of the Kunz polyhedron

Question

When are 2 numerical semigroups in the relative interior of the same face?

Christopher O’Neill (SDSU) Enumerating numerical semigroups May 4, 2019 14 / 18

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SLIDE 79

Faces of the Kunz polyhedron

Question

When are 2 numerical semigroups in the relative interior of the same face?

Christopher O’Neill (SDSU) Enumerating numerical semigroups May 4, 2019 14 / 18

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SLIDE 80

Faces of the Kunz polyhedron

Question

When are 2 numerical semigroups in the relative interior of the same face?

Christopher O’Neill (SDSU) Enumerating numerical semigroups May 4, 2019 14 / 18

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SLIDE 81

Faces of the Kunz polyhedron

Question

When are 2 numerical semigroups in the relative interior of the same face?

Christopher O’Neill (SDSU) Enumerating numerical semigroups May 4, 2019 14 / 18

slide-82
SLIDE 82

Faces of the Kunz polyhedron

Question

When are 2 numerical semigroups in the relative interior of the same face?

Christopher O’Neill (SDSU) Enumerating numerical semigroups May 4, 2019 14 / 18

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SLIDE 83

Faces of the Kunz polyhedron

Question

When are 2 numerical semigroups in the relative interior of the same face?

Christopher O’Neill (SDSU) Enumerating numerical semigroups May 4, 2019 14 / 18

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SLIDE 84

Verifying Wilf’s conjecture in fixed multiplicity

Wilf’s Conjecture

For any numerical semigroup S, F(S) + 1 ≤ e(S)(F(S) + 1 − g(S)).

Christopher O’Neill (SDSU) Enumerating numerical semigroups May 4, 2019 15 / 18

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SLIDE 85

Verifying Wilf’s conjecture in fixed multiplicity

Wilf’s Conjecture

For any numerical semigroup S, F(S) + 1 ≤ e(S)(F(S) + 1 − g(S)). Ingredients to Wilf’s inequality: g(S) = x1 + · · · + xm−1, linear in x1, . . . , xm−1;

Christopher O’Neill (SDSU) Enumerating numerical semigroups May 4, 2019 15 / 18

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SLIDE 86

Verifying Wilf’s conjecture in fixed multiplicity

Wilf’s Conjecture

For any numerical semigroup S, F(S) + 1 ≤ e(S)(F(S) + 1 − g(S)). Ingredients to Wilf’s inequality: g(S) = x1 + · · · + xm−1, linear in x1, . . . , xm−1; F(S) = max{mxi + i} − m, linear after checking some inequalities;

Christopher O’Neill (SDSU) Enumerating numerical semigroups May 4, 2019 15 / 18

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SLIDE 87

Verifying Wilf’s conjecture in fixed multiplicity

Wilf’s Conjecture

For any numerical semigroup S, F(S) + 1 ≤ e(S)(F(S) + 1 − g(S)). Ingredients to Wilf’s inequality: g(S) = x1 + · · · + xm−1, linear in x1, . . . , xm−1; F(S) = max{mxi + i} − m, linear after checking some inequalities; e(S), fixed on interior of each face.

Christopher O’Neill (SDSU) Enumerating numerical semigroups May 4, 2019 15 / 18

slide-88
SLIDE 88

Verifying Wilf’s conjecture in fixed multiplicity

Wilf’s Conjecture

For any numerical semigroup S, F(S) + 1 ≤ e(S)(F(S) + 1 − g(S)). Ingredients to Wilf’s inequality: g(S) = x1 + · · · + xm−1, linear in x1, . . . , xm−1; F(S) = max{mxi + i} − m, linear after checking some inequalities; e(S), fixed on interior of each face. The point: Wilf’s inequality is linear in the interior of a given face F ⊂ Pm.

Christopher O’Neill (SDSU) Enumerating numerical semigroups May 4, 2019 15 / 18

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SLIDE 89

Verifying Wilf’s conjecture in fixed multiplicity

Wilf’s Conjecture

For any numerical semigroup S, F(S) + 1 ≤ e(S)(F(S) + 1 − g(S)). Ingredients to Wilf’s inequality: g(S) = x1 + · · · + xm−1, linear in x1, . . . , xm−1; F(S) = max{mxi + i} − m, linear after checking some inequalities; e(S), fixed on interior of each face. The point: Wilf’s inequality is linear in the interior of a given face F ⊂ Pm. In particular, counterexamples to Wilf’s conjecture in the interior of F are precisely the set of integer solutions to a system of linear inequalities.

Christopher O’Neill (SDSU) Enumerating numerical semigroups May 4, 2019 15 / 18

slide-90
SLIDE 90

Verifying Wilf’s conjecture in fixed multiplicity

Wilf’s Conjecture

For any numerical semigroup S, F(S) + 1 ≤ e(S)(F(S) + 1 − g(S)). Ingredients to Wilf’s inequality: g(S) = x1 + · · · + xm−1, linear in x1, . . . , xm−1; F(S) = max{mxi + i} − m, linear after checking some inequalities; e(S), fixed on interior of each face. The point: Wilf’s inequality is linear in the interior of a given face F ⊂ Pm. In particular, counterexamples to Wilf’s conjecture in the interior of F are precisely the set of integer solutions to a system of linear inequalities. Example: S = 6, 9, 20 is a counterexample to Wilf’s conjecture iff

2x2 = x4 x2 + x3 = x5 x2 + x5 = x1 − 1 x3 + x4 = x1 − 1 2x1 > x2 x1 + x2 > x3 x1 + x3 > x4 x1 + x4 > x5 2x4 + 1 > x2 2x5 + 1 > x4 x3 + x5 + 1 > x2 x4 + x5 + 1 > x3 x1 − x2 ≥ 1 x1 − x3 ≥ 1 x1 − x4 ≥ 1 x1 − x5 ≥ 1 −11x1 + 3x2 + 3x3 + 3x4 + 3x5 > −7

Christopher O’Neill (SDSU) Enumerating numerical semigroups May 4, 2019 15 / 18

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SLIDE 91

Verifying Wilf’s conjecture in fixed multiplicity

Wilf’s Conjecture

For any numerical semigroup S, F(S) + 1 ≤ e(S)(F(S) + 1 − g(S)).

Christopher O’Neill (SDSU) Enumerating numerical semigroups May 4, 2019 16 / 18

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SLIDE 92

Verifying Wilf’s conjecture in fixed multiplicity

Wilf’s Conjecture

For any numerical semigroup S, F(S) + 1 ≤ e(S)(F(S) + 1 − g(S)). Algorithm for checking Wilf’s conjecture in multiplicity m: For each face F ⊂ Pm and each f ∈ [1, m − 1], search region

defining equalities for F, remaining inequalities for Pm (strict), Frobenius inequalities ensuring xf is maximal, and negation of Wilf’s inequality

for positive integer points. Any integer points found are counterexamples to Wilf’s conjecture.

Christopher O’Neill (SDSU) Enumerating numerical semigroups May 4, 2019 16 / 18

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SLIDE 93

Verifying Wilf’s conjecture in fixed multiplicity

Wilf’s Conjecture

For any numerical semigroup S, F(S) + 1 ≤ e(S)(F(S) + 1 − g(S)). Algorithm for checking Wilf’s conjecture in multiplicity m: For each face F ⊂ Pm and each f ∈ [1, m − 1], search region

defining equalities for F, remaining inequalities for Pm (strict), Frobenius inequalities ensuring xf is maximal, and negation of Wilf’s inequality

for positive integer points. Any integer points found are counterexamples to Wilf’s conjecture.

Theorem

Wilf’s conjecture holds for all numerical semigroups S with m(S) ≤ 18.

Christopher O’Neill (SDSU) Enumerating numerical semigroups May 4, 2019 16 / 18

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SLIDE 94

Runtimes

m # ineqs # extremal rays faces total time ≈ RAM 11 50 812 155,944 0.7 s 6 MB 12 60 1,864 669,794 2.5 s 35 MB 13 72 7,005 4,389,234 23 s 80 MB 14 84 15,585 21,038,016 1:19 m 603 MB 15 98 67,262 137,672,474 19:43 m 2.6 GB 16 112 184,025 751,497,188 1:35 h 12 GB 17 128 851,890 5,342,388,604 38:46 h 48 GB 18 144 2,158,379 28,275,375,292 29:05 d 720 GB 19 162 11,665,781 ?? ?? ??

Christopher O’Neill (SDSU) Enumerating numerical semigroups May 4, 2019 17 / 18

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SLIDE 95

References

  • W. Bruns, P. Garc´

ıa-S´ anchez, C. O’Neill, and D. Wilburne (2019) Wilf’s conjecture in fixed multiplicity preprint, available at [arXiv:1903.04342]

  • W. Bruns, B. Ichim, T. R¨
  • mer, R. Sieg and C. S¨
  • ger (2019)

Normaliz: algorithms for rational cones and affine monoids available at http://normaliz.uos.de

  • M. Delgado (2019)

Conjecture of Wilf: a survey preprint, available at [arXiv:1902.03461]

  • E. Kunz (1987)

¨ Uber die Klassifikation numerischer Halbgruppen Regensburger Mathematische Schriften 11, 1987.

  • H. Wilf (1978)

A circle-of-lights algorithm for the money-changing problem American Mathematics Monthly, 85 (1978) 562–565.

Christopher O’Neill (SDSU) Enumerating numerical semigroups May 4, 2019 18 / 18

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SLIDE 96

References

  • W. Bruns, P. Garc´

ıa-S´ anchez, C. O’Neill, and D. Wilburne (2019) Wilf’s conjecture in fixed multiplicity preprint, available at [arXiv:1903.04342]

  • W. Bruns, B. Ichim, T. R¨
  • mer, R. Sieg and C. S¨
  • ger (2019)

Normaliz: algorithms for rational cones and affine monoids available at http://normaliz.uos.de

  • M. Delgado (2019)

Conjecture of Wilf: a survey preprint, available at [arXiv:1902.03461]

  • E. Kunz (1987)

¨ Uber die Klassifikation numerischer Halbgruppen Regensburger Mathematische Schriften 11, 1987.

  • H. Wilf (1978)

A circle-of-lights algorithm for the money-changing problem American Mathematics Monthly, 85 (1978) 562–565. Thanks!

Christopher O’Neill (SDSU) Enumerating numerical semigroups May 4, 2019 18 / 18