Minimal presentations of shifted numerical monoids Christopher - PowerPoint PPT Presentation

Minimal presentations of shifted numerical monoids Christopher O’Neill Texas A&M University coneill@math.tamu.edu Joint with Rebecca Conaway*, Felix Gotti, Jesse Horton*, Roberto Pelayo, Mesa Williams*, and Brian Wissman * = undergraduate student May 7, 2016 Christopher O’Neill (Texas A&M University) Shifted numerical monoids May 7, 2016 1 / 15

Numerical monoids Definition A numerical monoid S is an additive submonoid of N with | N \ S | < ∞ . Christopher O’Neill (Texas A&M University) Shifted numerical monoids May 7, 2016 2 / 15

Numerical monoids Definition A numerical monoid S is an additive submonoid of N with | N \ S | < ∞ . Example McN = � 6 , 9 , 20 � = { 0 , 6 , 9 , 12 , 15 , 18 , 20 , 21 , . . . } . Christopher O’Neill (Texas A&M University) Shifted numerical monoids May 7, 2016 2 / 15

Numerical monoids Definition A numerical monoid S is an additive submonoid of N with | N \ S | < ∞ . Example McN = � 6 , 9 , 20 � = { 0 , 6 , 9 , 12 , 15 , 18 , 20 , 21 , . . . } . “McNugget Monoid” Christopher O’Neill (Texas A&M University) Shifted numerical monoids May 7, 2016 2 / 15

Numerical monoids Definition A numerical monoid S is an additive submonoid of N with | N \ S | < ∞ . Example McN = � 6 , 9 , 20 � = { 0 , 6 , 9 , 12 , 15 , 18 , 20 , 21 , . . . } . “McNugget Monoid” Factorizations: 60 = Christopher O’Neill (Texas A&M University) Shifted numerical monoids May 7, 2016 2 / 15

Numerical monoids Definition A numerical monoid S is an additive submonoid of N with | N \ S | < ∞ . Example McN = � 6 , 9 , 20 � = { 0 , 6 , 9 , 12 , 15 , 18 , 20 , 21 , . . . } . “McNugget Monoid” Factorizations: 60 = 7(6) + 2(9) Christopher O’Neill (Texas A&M University) Shifted numerical monoids May 7, 2016 2 / 15

Numerical monoids Definition A numerical monoid S is an additive submonoid of N with | N \ S | < ∞ . Example McN = � 6 , 9 , 20 � = { 0 , 6 , 9 , 12 , 15 , 18 , 20 , 21 , . . . } . “McNugget Monoid” Factorizations: 60 = 7(6) + 2(9) = 3(20) Christopher O’Neill (Texas A&M University) Shifted numerical monoids May 7, 2016 2 / 15

Numerical monoids Definition A numerical monoid S is an additive submonoid of N with | N \ S | < ∞ . Example McN = � 6 , 9 , 20 � = { 0 , 6 , 9 , 12 , 15 , 18 , 20 , 21 , . . . } . “McNugget Monoid” Factorizations: 60 = 7(6) + 2(9) � (7 , 2 , 0) = 3(20) (0 , 0 , 3) � Christopher O’Neill (Texas A&M University) Shifted numerical monoids May 7, 2016 2 / 15

50 100 150 200 250 0 5 10 15 To shift a numerical monoid. . . Fix S = � r 1 , . . . , r k � ⊂ ( N , +), and let M n = � n , n + r 1 , . . . , n + r k � . Christopher O’Neill (Texas A&M University) Shifted numerical monoids May 7, 2016 3 / 15

250 0 15 10 50 100 150 200 5 To shift a numerical monoid. . . Fix S = � r 1 , . . . , r k � ⊂ ( N , +), and let M n = � n , n + r 1 , . . . , n + r k � . Catenary degree c( M n ): measures spread of factorizations in M n . Christopher O’Neill (Texas A&M University) Shifted numerical monoids May 7, 2016 3 / 15

250 0 15 10 5 50 100 150 200 To shift a numerical monoid. . . Fix S = � r 1 , . . . , r k � ⊂ ( N , +), and let M n = � n , n + r 1 , . . . , n + r k � . Catenary degree c( M n ): measures spread of factorizations in M n . M n = � n , n + 6 , n + 9 , n + 20 � : Christopher O’Neill (Texas A&M University) Shifted numerical monoids May 7, 2016 3 / 15

250 0 15 10 5 50 100 150 200 To shift a numerical monoid. . . Fix S = � r 1 , . . . , r k � ⊂ ( N , +), and let M n = � n , n + r 1 , . . . , n + r k � . Catenary degree c( M n ): measures spread of factorizations in M n . M n = � n , n + 6 , n + 9 , n + 20 � : Christopher O’Neill (Texas A&M University) Shifted numerical monoids May 7, 2016 3 / 15

200 250 15 10 5 0 50 100 150 To shift a numerical monoid. . . Fix S = � r 1 , . . . , r k � ⊂ ( N , +), and let M n = � n , n + r 1 , . . . , n + r k � . Catenary degree c( M n ): measures spread of factorizations in M n . M n = � n , n + 6 , n + 9 , n + 20 � : c( M n ) is periodic-linear (quasilinear) for n ≥ 126. Christopher O’Neill (Texas A&M University) Shifted numerical monoids May 7, 2016 3 / 15

To shift a numerical monoid. . . Fix S = � r 1 , . . . , r k � ⊂ ( N , +), and let M n = � n , n + r 1 , . . . , n + r k � . Christopher O’Neill (Texas A&M University) Shifted numerical monoids May 7, 2016 4 / 15

To shift a numerical monoid. . . Fix S = � r 1 , . . . , r k � ⊂ ( N , +), and let M n = � n , n + r 1 , . . . , n + r k � . Delta set ∆( M n ): successive factorization length differences in M n . Christopher O’Neill (Texas A&M University) Shifted numerical monoids May 7, 2016 4 / 15

To shift a numerical monoid. . . Fix S = � r 1 , . . . , r k � ⊂ ( N , +), and let M n = � n , n + r 1 , . . . , n + r k � . Delta set ∆( M n ): successive factorization length differences in M n . Theorem (Chapman-Kaplan-Lemburg-Niles-Zlogar, 2014) The delta set ∆( M n ) is singleton for n ≫ 0 . Christopher O’Neill (Texas A&M University) Shifted numerical monoids May 7, 2016 4 / 15

To shift a numerical monoid. . . Fix S = � r 1 , . . . , r k � ⊂ ( N , +), and let M n = � n , n + r 1 , . . . , n + r k � . Delta set ∆( M n ): successive factorization length differences in M n . Theorem (Chapman-Kaplan-Lemburg-Niles-Zlogar, 2014) The delta set ∆( M n ) is singleton for n ≫ 0 . M n = � n , n + 6 , n + 9 , n + 20 � : Christopher O’Neill (Texas A&M University) Shifted numerical monoids May 7, 2016 4 / 15

To shift a numerical monoid. . . Fix S = � r 1 , . . . , r k � ⊂ ( N , +), and let M n = � n , n + r 1 , . . . , n + r k � . Delta set ∆( M n ): successive factorization length differences in M n . Theorem (Chapman-Kaplan-Lemburg-Niles-Zlogar, 2014) The delta set ∆( M n ) is singleton for n ≫ 0 . M n = � n , n + 6 , n + 9 , n + 20 � : ∆( M n ) = { 1 } for all n ≥ 48 Christopher O’Neill (Texas A&M University) Shifted numerical monoids May 7, 2016 4 / 15

0 2000 4000 50 100 150 200 250 3000 1000 To shift a numerical monoid. . . Fix S = � r 1 , . . . , r k � ⊂ ( N , +), and let M n = � n , n + r 1 , . . . , n + r k � . Christopher O’Neill (Texas A&M University) Shifted numerical monoids May 7, 2016 5 / 15

0 1000 4000 3000 50 100 150 200 250 2000 To shift a numerical monoid. . . Fix S = � r 1 , . . . , r k � ⊂ ( N , +), and let M n = � n , n + r 1 , . . . , n + r k � . Betti numbers β i ( M n ): Betti numbers of the defining toric ideal I M n . Christopher O’Neill (Texas A&M University) Shifted numerical monoids May 7, 2016 5 / 15

250 0 4000 3000 2000 1000 50 100 150 200 To shift a numerical monoid. . . Fix S = � r 1 , . . . , r k � ⊂ ( N , +), and let M n = � n , n + r 1 , . . . , n + r k � . Betti numbers β i ( M n ): Betti numbers of the defining toric ideal I M n . Theorem (Vu, 2014) The Betti numbers of M n are eventually r k -periodic in n. Christopher O’Neill (Texas A&M University) Shifted numerical monoids May 7, 2016 5 / 15

200 250 4000 3000 2000 1000 0 50 100 150 To shift a numerical monoid. . . Fix S = � r 1 , . . . , r k � ⊂ ( N , +), and let M n = � n , n + r 1 , . . . , n + r k � . Betti numbers β i ( M n ): Betti numbers of the defining toric ideal I M n . Theorem (Vu, 2014) The Betti numbers of M n are eventually r k -periodic in n. M n = � n , n + 6 , n + 9 , n + 20 � : Graded degrees for β 0 ( M n ) Christopher O’Neill (Texas A&M University) Shifted numerical monoids May 7, 2016 5 / 15

To shift a numerical monoid. . . Fix S = � r 1 , . . . , r k � ⊂ ( N , +), and let M n = � n , n + r 1 , . . . , n + r k � . Christopher O’Neill (Texas A&M University) Shifted numerical monoids May 7, 2016 6 / 15

To shift a numerical monoid. . . Fix S = � r 1 , . . . , r k � ⊂ ( N , +), and let M n = � n , n + r 1 , . . . , n + r k � . Observations: Christopher O’Neill (Texas A&M University) Shifted numerical monoids May 7, 2016 6 / 15

To shift a numerical monoid. . . Fix S = � r 1 , . . . , r k � ⊂ ( N , +), and let M n = � n , n + r 1 , . . . , n + r k � . Observations: Known: the Betti numbers n �→ β i ( M n ) are eventually r k -periodic. Christopher O’Neill (Texas A&M University) Shifted numerical monoids May 7, 2016 6 / 15

To shift a numerical monoid. . . Fix S = � r 1 , . . . , r k � ⊂ ( N , +), and let M n = � n , n + r 1 , . . . , n + r k � . Observations: Known: the Betti numbers n �→ β i ( M n ) are eventually r k -periodic. Known: the function n �→ ∆( M n ) is eventually singleton. Christopher O’Neill (Texas A&M University) Shifted numerical monoids May 7, 2016 6 / 15

To shift a numerical monoid. . . Fix S = � r 1 , . . . , r k � ⊂ ( N , +), and let M n = � n , n + r 1 , . . . , n + r k � . Observations: Known: the Betti numbers n �→ β i ( M n ) are eventually r k -periodic. Known: the function n �→ ∆( M n ) is eventually singleton. Observed: the function n �→ c( M n ) is eventually r k -quasilinear. Christopher O’Neill (Texas A&M University) Shifted numerical monoids May 7, 2016 6 / 15