Augmented Hilbert series of numerical semigroups Christopher O’Neill University of California Davis coneill@math.ucdavis.edu Joint with *Jeske Glenn, Vadim Ponomarenko, and *Benjamin Sepanski. * = undergraduate student January 12, 2018 Christopher O’Neill (UC Davis) Augmented Hilbert series January 12, 2018 1 / 16
Numerical semigroups Definition A numerical semigroup S ⊂ N : additive submsemigroup, | N \ S | < ∞ . Christopher O’Neill (UC Davis) Augmented Hilbert series January 12, 2018 2 / 16
Numerical semigroups Definition A numerical semigroup S ⊂ N : additive submsemigroup, | N \ S | < ∞ . Example McN = � 6 , 9 , 20 � = { 0 , 6 , 9 , 12 , 15 , 18 , 20 , . . . } . Christopher O’Neill (UC Davis) Augmented Hilbert series January 12, 2018 2 / 16
Numerical semigroups Definition A numerical semigroup S ⊂ N : additive submsemigroup, | N \ S | < ∞ . Example McN = � 6 , 9 , 20 � = { 0 , 6 , 9 , 12 , 15 , 18 , 20 , . . . } . “McNugget Semigroup” Christopher O’Neill (UC Davis) Augmented Hilbert series January 12, 2018 2 / 16
Numerical semigroups Definition A numerical semigroup S ⊂ N : additive submsemigroup, | N \ S | < ∞ . Example McN = � 6 , 9 , 20 � = { 0 , 6 , 9 , 12 , 15 , 18 , 20 , . . . } . “McNugget Semigroup” Factorizations: 60 = Christopher O’Neill (UC Davis) Augmented Hilbert series January 12, 2018 2 / 16
Numerical semigroups Definition A numerical semigroup S ⊂ N : additive submsemigroup, | N \ S | < ∞ . Example McN = � 6 , 9 , 20 � = { 0 , 6 , 9 , 12 , 15 , 18 , 20 , . . . } . “McNugget Semigroup” Factorizations: 60 = 7(6) + 2(9) Christopher O’Neill (UC Davis) Augmented Hilbert series January 12, 2018 2 / 16
Numerical semigroups Definition A numerical semigroup S ⊂ N : additive submsemigroup, | N \ S | < ∞ . 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 (UC Davis) Augmented Hilbert series January 12, 2018 2 / 16
Numerical semigroups Definition A numerical semigroup S ⊂ N : additive submsemigroup, | N \ S | < ∞ . Example McN = � 6 , 9 , 20 � = { 0 , 6 , 9 , 12 , 15 , 18 , 20 , . . . } . “McNugget Semigroup” Factorizations: 60 = 7(6) + 2(9) (7 , 2 , 0) � = 3(20) (0 , 0 , 3) � Christopher O’Neill (UC Davis) Augmented Hilbert series January 12, 2018 2 / 16
Numerical semigroups Fix a numerical semigroup S = � n 1 , . . . , n k � . � � a ∈ N k : n = a 1 n 1 + · · · + a k n k Z( n ) = is the set of factorizations of n ∈ S . Christopher O’Neill (UC Davis) Augmented Hilbert series January 12, 2018 3 / 16
Numerical semigroups Fix a numerical semigroup S = � n 1 , . . . , n k � . � � a ∈ N k : n = a 1 n 1 + · · · + a k n k Z( n ) = is the set of factorizations of n ∈ S . | a | = a 1 + · · · + a k ( length of a ) Christopher O’Neill (UC Davis) Augmented Hilbert series January 12, 2018 3 / 16
Numerical semigroups Fix a numerical semigroup S = � n 1 , . . . , n k � . � � a ∈ N k : n = a 1 n 1 + · · · + a k n k Z( n ) = is the set of factorizations of n ∈ S . | a | = a 1 + · · · + a k ( 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 (UC Davis) Augmented Hilbert series January 12, 2018 3 / 16
Numerical semigroups Fix a numerical semigroup S = � n 1 , . . . , n k � . � � a ∈ N k : n = a 1 n 1 + · · · + a k n k Z( n ) = is the set of factorizations of n ∈ S . | a | = a 1 + · · · + a k ( 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 (UC Davis) Augmented Hilbert series January 12, 2018 3 / 16
Numerical semigroups Fix a numerical semigroup S = � n 1 , . . . , n k � . � � a ∈ N k : n = a 1 n 1 + · · · + a k n k Z( n ) = is the set of factorizations of n ∈ S . | a | = a 1 + · · · + a k ( 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 (UC Davis) Augmented Hilbert series January 12, 2018 3 / 16
Numerical semigroups Fix a numerical semigroup S = � n 1 , . . . , n k � . � � a ∈ N k : n = a 1 n 1 + · · · + a k n k Z( n ) = is the set of factorizations of n ∈ S . | a | = a 1 + · · · + a k ( 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 (UC Davis) Augmented Hilbert series January 12, 2018 3 / 16
Numerical semigroups Fix a numerical semigroup S = � n 1 , . . . , n k � . � � a ∈ N k : n = a 1 n 1 + · · · + a k n k Z( n ) = is the set of factorizations of n ∈ S . | a | = a 1 + · · · + a k ( 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 (UC Davis) Augmented Hilbert series January 12, 2018 3 / 16
Numerical semigroups Fix a numerical semigroup S = � n 1 , . . . , n k � . � � a ∈ N k : n = a 1 n 1 + · · · + a k n k Z( n ) = is the set of factorizations of n ∈ S . | a | = a 1 + · · · + a k ( 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) (166662 , 1 , 1) } , . . . , � �� � � �� � shortest longest Christopher O’Neill (UC Davis) Augmented Hilbert series January 12, 2018 3 / 16
Maximum and minimum factorization length Let S = � n 1 , . . . , n k � . For n ∈ S , M( n ) = max length m( n ) = min length Christopher O’Neill (UC Davis) Augmented Hilbert series January 12, 2018 4 / 16
Maximum and minimum factorization length Let S = � n 1 , . . . , n k � . For n ∈ S , M( n ) = max length m( n ) = min length Observations Christopher O’Neill (UC Davis) Augmented Hilbert series January 12, 2018 4 / 16
Maximum and minimum factorization length Let S = � n 1 , . . . , n k � . For n ∈ S , M( n ) = max length m( n ) = min length Observations Max length factorization: lots of small generators Christopher O’Neill (UC Davis) Augmented Hilbert series January 12, 2018 4 / 16
Maximum and minimum factorization length Let S = � n 1 , . . . , n k � . For n ∈ S , M( n ) = max length m( n ) = min length Observations Max length factorization: lots of small generators Min length factorization: lots of large generators Christopher O’Neill (UC Davis) Augmented Hilbert series January 12, 2018 4 / 16
Maximum and minimum factorization length Let S = � n 1 , . . . , n k � . For n ∈ S , M( n ) = max length m( n ) = min length Observations Max length factorization: lots of small generators Min length factorization: lots of large generators Example S = � 6 , 9 , 20 � : Christopher O’Neill (UC Davis) Augmented Hilbert series January 12, 2018 4 / 16
Maximum and minimum factorization length Let S = � n 1 , . . . , n k � . For n ∈ S , M( n ) = max length m( n ) = min length Observations Max length factorization: lots of small generators Min length factorization: lots of large generators Example S = � 6 , 9 , 20 � : M(40) = 2 and Z(40) = { (0 , 0 , 2) } Christopher O’Neill (UC Davis) Augmented Hilbert series January 12, 2018 4 / 16
Maximum and minimum factorization length Let S = � n 1 , . . . , n k � . For n ∈ S , M( n ) = max length m( n ) = min length Observations Max length factorization: lots of small generators Min length factorization: lots of large generators Example S = � 6 , 9 , 20 � : M(40) = 2 and Z(40) = { (0 , 0 , 2) } S = � 5 , 16 , 17 , 18 , 19 � : Christopher O’Neill (UC Davis) Augmented Hilbert series January 12, 2018 4 / 16
Maximum and minimum factorization length Let S = � n 1 , . . . , n k � . For n ∈ S , M( n ) = max length m( n ) = min length Observations Max length factorization: lots of small generators Min length factorization: lots of large generators Example S = � 6 , 9 , 20 � : M(40) = 2 and Z(40) = { (0 , 0 , 2) } S = � 5 , 16 , 17 , 18 , 19 � : m(82) = 5 and Z(82) = { (0 , 3 , 2 , 0 , 0) , (5 , 0 , 0 , 0 , 3) , . . . } Christopher O’Neill (UC Davis) Augmented Hilbert series January 12, 2018 4 / 16
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