Delta Set of Affine Semigroups

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Basil Stone
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1 of Affine Semigroups Carrasco Departament of Mathematics University Of Almería Conference on Rings and Factorizations Graz 2018

2 2 / 20 s This is a work collecting some ideas published in the next three papers: García-Sánchez P.A., Ll. D., Moscariello A. Delta sets for nonsymmetric numerical semigroups with dimension three. Forum Math. 30 pp (2018). García-Sánchez P.A., Ll. D., Moscariello A. Delta sets for symmetric numerical semigroups with dimension three. Aequat. Math. 91 pp (2017). García-Sánchez P.A., O Neill Ch., Webb G. On the computation of factorization invariants for affine semigroups J. Algebra & Appl. to appear.

3 3 / 20 s Congruences and minimal presentations S = s 1,..., s n N d is an affine semigroup generated by s 1,..., s n if S = {z 1 s z n s n z 1,..., z n N {0}}. When d = 1 and gcd(s 1,..., s n ) = 1 we say a numerical semigroup. ϕ S : N n S ; ϕ(z 1,..., z n ) = z 1 s z n s n is surjective and S N n / ker ϕ S, where ker ϕ S = {(z, z ) N n N n : ϕ S (z) = ϕ S (z )}. ker ϕ S is a congruence on N n.

4 4 / 20 s Congruences and minimal presentations ker ϕ S Is finitely generated and a subset ρ ker ϕ S that generates ker ϕ which is minimal w.r.t. containment is called minimal presentation of S. This minimal presentation need not be unique Example 1: S = 3, 5, 7 a numerical semigroup ρ = {((1, 0, 1), (0, 2, 0)), ((4, 0, 0), (0, 1, 1)), ((3, 1, 0), (0, 0, 2))} In this case ρ is unique. Example 2: S = 4, 6, 11 a numerical semigroup ρ = {((1, 3, 0), (0, 0, 2)), ((3, 0, 0), (0, 2, 0))}. ρ = {((4, 1, 0), (0, 0, 2)), ((3, 0, 0), (0, 2, 0))} In this case, there exist several minimal presentations.

5 5 / 20 s S an affine semigroup The Betti elements of S are the elements ϕ S (z) for (z, z ) ρ This definition is independent of the minimal presentation chosen Examples S = 3, 5, 7 ; then Betti(S ) = {10, 12, 14} from ρ = {((1, 0, 1), (0, 2, 0)), ((4, 0, 0), (0, 1, 1)), ((3, 1, 0), (0, 0, 2))}. S = 4, 6, 11 ; then Betti(S ) = {12, 22} from ρ = {((1, 3, 0), (0, 0, 2)), ((3, 0, 0), (0, 2, 0))} or ρ = {((4, 1, 0), (0, 0, 2)), ((3, 0, 0), (0, 2, 0))}.

6 6 / 20 s Factorizations Factorizations of an element s S = s 1,..., s n Z(s) = {(z 1,..., z n ) N n, with s = z 1 s z n s n } Length of a factorization z = (z 1,..., z n ) l(z) = z z n Set of lengths of factorizations of s S Betti elements L(s) = {l(z) z Z(s)}. For each s S Betti elements are those whose factorizations are used to build minimal presentations

7 7 / 20 s S =< 3, 5, 7 >= {0, 3, 5, 6, 7, 8, 9, 10, 11,...} Elements 0, 3, 5, 6, 7, 8, 9, 11, have only one factorization. But: Z(10) = {(1, 0, 1), (0, 2, 0)} L(10) = {2} Z(12) = {(4, 0, 0), (0, 1, 1)} L(12) = {2, 4} Z(14) = {(3, 1, 0), (0, 0, 2)} L(14) = {2, 4} Z(30) = {(10, 0, 0), (5, 3, 0), (0, 6, 0), (6, 1, 1), (1, 4, 1), (2, 2, 2), (3, 0, 3)} L(30) = {6, 8, 10} We order the set L(s) which always is finite L(s) = {l 1 < l 2 < < l n } And define the Delta sets as: (s) = {l i l i 1 i = 2,... n}. (S ) = s S (s). We going to focus in the set (S ).

8 8 / 20 s Factorization and s Geroldinger Let S be a numerical semigroup, then: min (S ) = gcd (S ). Set d = gcd (S ). There exists k N \ {0} such that (S ) {d, 2d,..., kd}. Chapman,García-Sánchez,Ll,Malyshev,Steinberg For ρ a minimal presentation of S we have: min (S ) = gcd{ l(z) l(z ) : (z, z ) ρ} max (S ) = max{max (s): s Betti(S )}.

9 9 / 20 s Semigroups dimension 3 The M S group associated to S = s 1, s 2, s 3 M S = {(x 1, x 2, x 3 ) Z 3 x 1 s 1 + x 2 s 2 + x 3 s 3 = 0}. v 1 = (4, 1, 1) and v 2 = (3, 1, 2) span M S as a group. δ 1 = l(v 1 ) = 2 and δ 2 = l(v 2 ) = 2. And (S ) = {2}. For δ 1 and δ 2 coprime integer, define η 1 = max{δ 1, δ 2 }, η 2 = min{δ 1, δ 2 } and η i+2 = η i mod η i+1. Euclid s algorithm. The Euclid s set, Euc(δ 1, δ 2 ), can be constructed as D(η 1, η 2 ) = {η 1, η 1 η 2,..., η 1 mod η 2 = η 3 }, D(η 2, η 3 ) = {η 2, η 2 η 3,..., η 2 mod η 3 = η 4 }, D(η 3, η 4 ) = {η 3 η 4,..., η 3 mod η 4 = η 5 }, D(η i, η i+1 ) = {η i η i+1,..., η i mod η i+1 = η i+2 = 0}. Then: Euc(δ 1, δ 2 ) = D(η i, η i+1 ). i I

10 10 / 20 s Semigroups dimension 3 Theorem For S = n 1, n 2, n 3 we have: (S ) = Euc(δ 1, δ 2 ) In this case Betti(S ) = 1, 2 or 3 For Betti(S ) = 3. ρ is unique. And all elements have 2 factorizations. Then: v 1 = (c 1, r 12, r 13 ) and v 2 = (r 31, r 32, c 3 ). For Betti(S ) = 2. ρ is not unique. But we can choose: v 1 = (m 2, m 1, 0) and v 2 = (b + λm 2, c λm 1, a). For Betti(S ) = 1. ρ is not unique. But we can choose: v 1 = (s 1, s 2, 0) and v 2 = (0, s 2, s 3 ).

11 11 / 20 s Semigroups dimension 3 Example: S = 1407, 26962, v 1 = (411, 7, 11), v 2 = (127, 84, 69), and so: δ 1 = 393, δ 2 = 142. v 1 =(411, 7, 11) v 1 v 2 =(284, 91,58) v 1 2v 2 =(157, 175,127)=v = η 3 v 2 =(127,84, 69) v 2 v 3 =( 30,259, 196)=v = η 4 v 3 =(157, 175,127) (187, 434,323) (217, 693,519) (247, 952,715) = η 5 ( 30,259, 196) ( 277,1211, 911) ( 524,2163, 1626) ( 771,3115, 2341) = η 6 (247, 952,715) (1018, 4067,3056) (1789, 7182,5397) (2560, 10297,7738) = η 7 ( 771,3115, 2341) ( 3331,13412, 10079) ( 5891,23709, 17817) ( 8451,34006, 25555) = η 8 (S ) = {1, 2, 3, 4, 7, 10, 13, 23, 33, 43, 76, 109, 142, 251, 393}.

12 12 / 20 s Higher dimensions in numerical semigroups Colton and Kaplan example They show, in this example, that we can not generalize the results to higher dimensions. S = 14, 29, 30, 32, 36 (S ) = {1, 4} If we try to extrapolate our results, necessarily {2, 3} must be contained in the Delta set of 14, 29, 30, 32, 36.

13 13 / 20 s A Generalization to affine semigroups O Neill Ch.(2017) A first Theorem: For S = s 1,..., s n, consider the ideals: I j = y z y z : (z, z ) Z(s), s S, and l(z) l(z ) j K[y 1,..., y n ] Then j (S ) if and only if I j 1 I j strictly. We denote y z for the monomial y z 1 1 yz n n The trick: homogenize The new variable stores lengths, and splits factorizations in layers of factorizations with the same length For S H = (1, 0), (1, s 1 ),..., (1, s n ) N d+1 consider I S H = t l(z) l(z ) y z y z : z, z Z(s), s S, l(z) l(z ) K[t, y 1,..., y n ] For G a reduced Groebner basis of I S H with some order satisfying t > y i then J j = y z y z : t i y z y z G, i j = I j

14 14 / 20 s Algorithm function DeltaSetOfAffineSemigroup(S ) ρ minimal presentation of S H G reduced lex Groebner basis for t i y z t j y z : ((i, z), ( j, z )) ρ return { j: t j y z y z G} Return to S = 1407, 26962, ρ = {((411; 411, 0, 0), (18; 0, 7, 11)), ((211; 127, 84, 0), (69; 0, 0, 69)), ((342; 284, 0, 58), (91; 0, 91, 0))}. t 411 x 411 t 18 y 7 z 11, t 211 x 127 y 84 t 69 z 69, t 342 x 284 z 58 t 91 y 91, with t > x > y > z. t 393 x 411 y 7 z 11, t 142 x 127 y 84 z 69, t 251 x 284 z 58 y 91 = f 1, f 2, f 3 S ( f 1, f 2 ) = t393 x 411 y 84 t 393 x 411 f 1 t393 x 411 y 84 t 142 x 127 y 84 f 2 = t 251 x 284 z 69 y 91 z 11 t 251 x 284 z 58 y 91 = f 3 S ( f 3, f 2 ) = t251 x 284 y 84 z 58 t 251 x 284 z 58 f 3 t251 x 284 y 84 z 58 t 142 x 127 y 84 f 2 = t 109 x 157 z 127 y 175 = f 4 S ( f 2, f 4 ) = t251 x 284 z 127 t 109 x 157 z 58 f 2 t251 x 284 y 84 z 58 t 142 x 127 y 84 f 4 = t 109 x 157 z 127 y 175 = f 4

15 15 / 20 s Algorithm function DeltaSetOfAffineSemigroup(S ) ρ minimal presentation of S H G reduced lex Groebner basis for t i y z t j y z : ((i, z), ( j, z )) ρ return { j: t j y z y z G} Return to S = 1407, 26962, ρ = {((411; 411, 0, 0), (18; 0, 7, 11)), ((211; 127, 84, 0), (69; 0, 0, 69)), ((342; 284, 0, 58), (91; 0, 91, 0))}. t 411 x 411 t 18 y 7 z 11, t 211 x 127 y 84 t 69 z 69, t 342 x 284 z 58 t 91 y 91, with t > x > y > z. t 393 x 411 y 7 z 11, t 142 x 127 y 84 z 69, t 251 x 284 z 58 y 91 = f 1, f 2, f 3. S ( f 1, f 2 ) = t393 x 411 y 84 t 393 x 411 f 1 t393 x 411 y 84 t 142 x 127 y 84 f 2 = t 251 x 284 z 69 y 91 z 11 t 251 x 284 z 58 y 91 = f 3 S ( f 3, f 2 ) = t251 x 284 y 84 z 58 t 251 x 284 z 58 f 3 t251 x 284 y 84 z 58 t 142 x 127 y 84 f 2 = t 109 x 157 z 127 y 175 = f 4 S ( f 2, f 4 ) = t251 x 284 z 127 t 109 x 157 z 58 f 2 t251 x 284 y 84 z 58 t 142 x 127 y 84 f 4 = t 109 x 157 z 127 y 175 = f 4

16 16 / 20 s Algorithm function DeltaSetOfAffineSemigroup(S ) ρ minimal presentation of S H G reduced lex Groebner basis for t i y z t j y z : ((i, z), ( j, z )) ρ return { j: t j y z y z G} Return to S = 1407, 26962, ρ = {((411; 411, 0, 0), (18; 0, 7, 11)), ((211; 127, 84, 0), (69; 0, 0, 69)), ((342; 284, 0, 58), (91; 0, 91, 0))}. t 411 x 411 t 18 y 7 z 11, t 211 x 127 y 84 t 69 z 69, t 342 x 284 z 58 t 91 y 91, with t > x > y > z. t 393 x 411 y 7 z 11, t 142 x 127 y 84 z 69, t 251 x 284 z 58 y 91 = f 1, f 2, f 3. S ( f 1, f 2 ) = t393 x 411 y 84 t 393 x 411 f 1 t393 x 411 y 84 t 142 x 127 y 84 f 2 = t 251 x 284 z 69 y 91 z 11 t 251 x 284 z 58 y 91 = f 3 S ( f 3, f 2 ) = t251 x 284 y 84 z 58 t 251 x 284 z 58 f 3 t251 x 284 y 84 z 58 t 142 x 127 y 84 f 2 = t 109 x 157 z 127 y 175 = f 4 S ( f 2, f 4 ) = t251 x 284 z 127 t 109 x 157 z 58 f 2 t251 x 284 y 84 z 58 t 142 x 127 y 84 f 4 = t 109 x 157 z 127 y 175 = f 4

17 17 / 20 s Algorithm function DeltaSetOfAffineSemigroup(S ) ρ minimal presentation of S H G reduced lex Groebner basis for t i y z t j y z : ((i, z), ( j, z )) ρ return { j: t j y z y z G} Return to S = 1407, 26962, ρ = {((411; 411, 0, 0), (18; 0, 7, 11)), ((211; 127, 84, 0), (69; 0, 0, 69)), ((342; 284, 0, 58), (91; 0, 91, 0))}. t 411 x 411 t 18 y 7 z 11, t 211 x 127 y 84 t 69 z 69, t 342 x 284 z 58 t 91 y 91, with t > x > y > z. t 393 x 411 y 7 z 11, t 142 x 127 y 84 z 69, t 251 x 284 z 58 y 91 = f 1, f 2, f 3. S ( f 1, f 2 ) = t393 x 411 y 84 t 393 x 411 f 1 t393 x 411 y 84 t 142 x 127 y 84 f 2 = t 251 x 284 z 69 y 91 z 11 t 251 x 284 z 58 y 91 = f 3 S ( f 3, f 2 ) = t251 x 284 y 84 z 58 t 251 x 284 z 58 f 3 t251 x 284 y 84 z 58 t 142 x 127 y 84 f 2 = t 109 x 157 z 127 y 175 = f 4 S ( f 2, f 4 ) = t142 x 157 y 84 z 127 t 142 x 127 y 84 f 2 t142 x 157 y 84 z 127 t 109 x 157 z 127 f 4 = t 33 y 259 x 30 z 196 = f 5

18 18 / 20 s Algorithm function DeltaSetOfAffineSemigroup(S ) ρ minimal presentation of S H G reduced lex Groebner basis for t i y z t j y z : ((i, z), ( j, z )) ρ return { j: t j y z y z G} Return to S = 1407, 26962, ρ = {((411; 411, 0, 0), (18; 0, 7, 11)), ((211; 127, 84, 0), (69; 0, 0, 69)), ((342; 284, 0, 58), (91; 0, 91, 0))}. t 411 x 411 t 18 y 7 z 11, t 211 x 127 y 84 t 69 z 69, t 342 x 284 z 58 t 91 y 91, with t > x > y > z. t 393 x 411 y 7 z 11, t 142 x 127 y 84 z 69, t 251 x 284 z 58 y 91 = f 1, f 2, f 3. S ( f 1, f 2 ) = t393 x 411 y 84 t 393 x 411 f 1 t393 x 411 y 84 t 142 x 127 y 84 f 2 = t 251 x 284 z 69 y 91 z 11 t 251 x 284 z 58 y 91 = f 3 S ( f 3, f 2 ) = t251 x 284 y 84 z 58 t 251 x 284 z 58 f 3 t251 x 284 y 84 z 58 t 142 x 127 y 84 f 2 = t 109 x 157 z 127 y 175 = f 4 S ( f 2, f 4 ) = t142 x 157 y 84 z 127 t 142 x 127 y 84 f 2 t142 x 157 y 84 z 127 t 109 x 157 z 127 f 4 = t 33 y 259 x 30 z 196 = f 5... And, so on

19 19 / 20 s Algorithm function DeltaSetOfAffineSemigroup(S ) ρ minimal presentation of S H G reduced lex Groebner basis for t i y z t j y z : ((i, z), ( j, z )) ρ return { j: t j y z y z G} Return to S = 1407, 26962, ρ = {((411; 411, 0, 0), (18; 0, 7, 11)), ((211; 127, 84, 0), (69; 0, 0, 69)), ((342; 284, 0, 58), (91; 0, 91, 0))}. t 411 x 411 t 18 y 7 z 11, t 211 x 127 y 84 t 69 z 69, t 342 x 284 z 58 t 91 y 91, with t > x > y > z. t 393 x 411 y 7 z 11, t 142 x 127 y 84 z 69, t 251 x 284 z 58 y 91 = f 1, f 2, f 3. S ( f 1, f 2 ) = t393 x 411 y 84 t 393 x 411 f 1 t393 x 411 y 84 t 142 x 127 y 84 f 2 = t 251 x 284 z 69 y 91 z 11 t 251 x 284 z 58 y 91 = f 3 S ( f 3, f 2 ) = t251 x 284 y 84 z 58 t 251 x 284 z 58 f 3 t251 x 284 y 84 z 58 t 142 x 127 y 84 f 2 = t 109 x 157 z 127 y 175 = f 4 S ( f 2, f 4 ) = t142 x 157 y 84 z 127 t 142 x 127 y 84 f 2 t142 x 157 y 84 z 127 t 109 x 157 z 127 f 4 = t 33 y 259 x 30 z 196 = f 5... And, so on

20 20 / 20 s Thanks for your attention!!

On intersections of complete intersection ideals

On intersections of complete intersection ideals Mircea Cimpoeaş I.M.A.R. mircea.cimpoeas@imar.ro joint work with Dumitru Stamate July 7, 2016, I.M.N.S., Levico Terme, Italy Overview 1 Introduction 2 Main