Enumerating numerical semigroups using polyhedral geometry - PowerPoint PPT Presentation

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

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

Numerical semigroups Definition A numerical semigroup S ⊂ Z ≥ 0 : closed under addition , | Z ≥ 0 \ S | < ∞ . Example: � � 0 , 6 , 9 , 12 , 15 , 18 , 20 , 21 , 24 , . . . McN = � 6 , 9 , 20 � = . . . , 36 , 38 , 39 , 40 , 41 , 42 , 44 → Christopher O’Neill (SDSU) Enumerating numerical semigroups May 4, 2019 2 / 18

Numerical semigroups Definition A numerical semigroup S ⊂ Z ≥ 0 : closed under addition , | Z ≥ 0 \ S | < ∞ . Example: “McNugget Semigroup” � � 0 , 6 , 9 , 12 , 15 , 18 , 20 , 21 , 24 , . . . McN = � 6 , 9 , 20 � = . . . , 36 , 38 , 39 , 40 , 41 , 42 , 44 → Christopher O’Neill (SDSU) Enumerating numerical semigroups May 4, 2019 2 / 18

Numerical semigroups Definition A numerical semigroup S ⊂ Z ≥ 0 : closed under addition , | Z ≥ 0 \ S | < ∞ . Example: “McNugget Semigroup” � � 0 , 6 , 9 , 12 , 15 , 18 , 20 , 21 , 24 , . . . McN = � 6 , 9 , 20 � = . . . , 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

Numerical semigroups Definition A numerical semigroup S ⊂ Z ≥ 0 : closed under addition , | Z ≥ 0 \ S | < ∞ . Example: “McNugget Semigroup” � � 0 , 6 , 9 , 12 , 15 , 18 , 20 , 21 , 24 , . . . McN = � 6 , 9 , 20 � = . . . , 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

Numerical semigroups Definition A numerical semigroup S ⊂ Z ≥ 0 : closed under addition , | Z ≥ 0 \ S | < ∞ . Example: “McNugget Semigroup” � � 0 , 6 , 9 , 12 , 15 , 18 , 20 , 21 , 24 , . . . McN = � 6 , 9 , 20 � = . . . , 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

Numerical semigroups Definition A numerical semigroup S ⊂ Z ≥ 0 : closed under addition , | Z ≥ 0 \ S | < ∞ . Example: “McNugget Semigroup” � � 0 , 6 , 9 , 12 , 15 , 18 , 20 , 21 , 24 , . . . McN = � 6 , 9 , 20 � = . . . , 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

Numerical semigroups Definition A numerical semigroup S ⊂ Z ≥ 0 : closed under addition , | Z ≥ 0 \ S | < ∞ . Example: “McNugget Semigroup” � � 0 , 6 , 9 , 12 , 15 , 18 , 20 , 21 , 24 , . . . McN = � 6 , 9 , 20 � = . . . , 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

Numerical semigroups Definition A numerical semigroup S ⊂ Z ≥ 0 : closed under addition , | Z ≥ 0 \ S | < ∞ . Example: “McNugget Semigroup” � � 0 , 6 , 9 , 12 , 15 , 18 , 20 , 21 , 24 , . . . McN = � 6 , 9 , 20 � = . . . , 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

Frobenius number Fix a numerical semigroup S = � n 1 , . . . , n k � . Christopher O’Neill (SDSU) Enumerating numerical semigroups May 4, 2019 3 / 18

Frobenius number Fix a numerical semigroup S = � n 1 , . . . , n k � . Definition F( S ) = max( N \ S ) is the Frobenius number of S . Christopher O’Neill (SDSU) Enumerating numerical semigroups May 4, 2019 3 / 18

Frobenius number Fix a numerical semigroup S = � n 1 , . . . , n k � . 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

Frobenius number Fix a numerical semigroup S = � n 1 , . . . , n k � . 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

Frobenius number Fix a numerical semigroup S = � n 1 , . . . , n k � . 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 = � n 1 , n 2 � , then F( S ) = n 1 n 2 − ( n 1 + n 2 ). Christopher O’Neill (SDSU) Enumerating numerical semigroups May 4, 2019 3 / 18

Frobenius number Fix a numerical semigroup S = � n 1 , . . . , n k � . 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 = � n 1 , n 2 � , then F( S ) = n 1 n 2 − ( n 1 + n 2 ). If S = � n 1 , n 2 , n 3 � , then there is a fast algorithm for F ( S ). Christopher O’Neill (SDSU) Enumerating numerical semigroups May 4, 2019 3 / 18

Frobenius number Fix a numerical semigroup S = � n 1 , . . . , n k � . 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 = � n 1 , n 2 � , then F( S ) = n 1 n 2 − ( n 1 + n 2 ). If S = � n 1 , n 2 , n 3 � , 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

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

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

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

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

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

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