Stratified sets of atoms Pedro A. Garca Snchez Universidad de - PowerPoint PPT Presentation

Stratified sets of atoms Pedro A. García Sánchez Universidad de Granada (member of the arQus alliance) Joint work with D. Llena and U. Krause Additive Combinatorics - CIRM 2020

Motivation Let be the set of nonnegative integer solutions of the Diophantine equation M ax + by = cz , with , and positive integers a b c N 3 The monoid is a (full) affine monoid contained in , and it is isomorphic to , the M N set of nonnegative integer solutions of ax + by ≡ 0 (mod c ) N 2 This new monoid is a (full) affine semigroup of 2

Motivation (II) There are many procedures to find the set of atoms (minimal generators) of 2 N = {( x , y ) ∈ N ∣ ax + by ≡ 0 (mod c )} Thus there are ways to parametrize the set of all nonnegative integer solutions of ax + by ≡ 0 (mod c ) The problem is that for a solution ( x , y ) there can be different expressions (factorizations) in terms of the atoms of A Elliott back in 1903 was concerned with the problem of parametrizing "uniquely" the set of solutions of ax + by = cz 3

Inspiration ( N , +) N Let be a numerical semigroup (a submonoid of with finite complement in ) S Let be a positive integer in ; define m S Ap( S , m ) = S ∖ ( m + S ) = { s ∈ S ∣ s − m  ∈ S } x ∈ S Then every can be expressed uniquely as x = km + w , k ∈ N with and w ∈ Ap( S , m ) 4

Inspiration (II) Q = If is a simplicial affine Cohen-Macaulay semigroup, with a set of extreme rays S { r , … , r } and set of atoms , then every element x ∈ S can be expressed A 1 n uniquely as x = λ r + ⋯ + λ r + w , 1 1 n n N with λ ∈ and w ∈ Ap( S , Q ) = S ∖ ( Q + S ) i S ∖ ( Q + S ) The set has finitely many elements; these are combinations of the elements in A ∖ Q 5

Some definitions Let a monoid. A submonoid of is a face of if x + y ∈ F with x , y ∈ M , M F M M forces x , y ∈ F N n Given a subset of , define the cone spanned by as X X k ∈ N , q ∈ Q L ( X ) = { q x + ⋯ + q x ∣ , x ∈ X } Q ≥0 1 1 ≥0 k k i i Q For an affine semigroup and an atom of , we say that is an exteme ray if S a S a a ≥0 is a face of L ( S ) Q ≥0 S ⊆ N n An affine semigroup is simplicial if there exist linearly independent extreme rays r , … , r such that L ( S ) = L ({ r , … , r }) Q ≥0 Q ≥0 1 1 n n An affine semigroup is Cohen-Macaulay if the semigroup ring K [ S ] is Cohen- S Macaulay (if is simplicial it does not depend on ) S K 6

Good news The monoid 2 N = {( x , y ) ∈ N ∣ ax + by ≡ 0 (mod c )} is simplicial, Cohen-Macaulay, the extreme rays and the Apéry set with respect to them are straightforward to compute Thus any element in can be expressed as N λ ( n , 0) + μ (0, m ) + w with ( n , 0), (0, m ) the extreme rays of and w ∈ Ap( N , {( n , 0), (0, m )}) N 7

Root-closed inside factorial monoids Our monoids are cancellative, atomic and reduced We say that a monoid is inside factorial if there exists a factorial (free) submonoid M N of such that for any m ∈ M there exists a positive integer with km ∈ N M k If is the set of atoms of , then we say that is inside factorial with base Q N M Q This concept generalizes that of simplicial affine semigroup We say that is root-closed if for every a , b ∈ M and a positive integer such that M n n ( a − b ) ∈ M , we have that a − b ∈ M Every full affine semigroup is root-closed 8

Uniqueness of expressions on root-closed inside factorial monoids Let be a root-closed inside factorial monoid with basis . Then M Q ⋃ M = a + ⟨ Q ⟩ a ∈Ap( M , Q ) and this union is disjoint In particular, is a disjoint union of translates of a factorial monoid M Also every element is written uniquely as a combination of elements in plus an Q element in Ap( M , Q ) = M ∖ Q + M 9

Structure - Abstraction Let be a root-closed inside factorial monoid with basis M Q a , b ∈ Ap( M , Q ) a + b = c + I ( a , b ) c ∈ Ap( M , Q ) Take . Then for some and I ( a , b ) ∈ ⟨ Q ⟩ a ⊕ b = c Ap( M , Q ) We write . With this operation becomes a torsion group, and I has some special properties In fact, every root-closed inside factorial monoid is isomorphic to G × F , with a G torsion group and a free monoid, endowed with the operation F ( a , f ) + I ( b , g ) = ( a + b , f + g + I ( a , b )) I : G × G → F with fulfilling the properties mentioned above 10

Extraction grades and Apéry sets Let be a monoid. The extraction grade for x , y ∈ M ∖ {0} is M λ ( x , y ) = sup{ m / n ∣ ny − mx ∈ M , m , n ∈ Z } + If is inside factorial with base , then M Q { x ∈ M ∣ λ ( q , x ) < 1 for all q ∈ Q } ⊆ Ap( M , Q ) Equality holds when is root-closed M So every element x ∈ M can be written uniquely as ∑ x = λ q + a q q ∈ Q N λ ( q , a ) < 1 q ∈ Q λ ∈ with for all (finite sum and ) q 11

Strong atoms N a An atom in an atomic monoid is strong if is a face of a M M N a ∩ N b = ∅ In a root-closed atomic monoid any two atoms are disjoint ( ) M If is inside factorial and root-closed, then the base is the set of strong atoms of M Q M If is simplicial, root-closed and affine, then its strong atoms are precisely the M extreme rays of M 12

The idea of stratification N 2 Assume that lives inside and that it is root-closed M Trivially, is simplicial (inside factorial) and has say Q = { q , q } as extreme rays M 1 2 Every x ∈ M will we expressed uniquely as x = λ q + λ q + a 1 1 2 2 λ ( q , a ) < 1 λ ( q , a ) < 1 with and 1 2 H = Q ∪ H ′ ′ ′ Assume that the atoms of are . Set H = and M = ⟨ H ⟩ M Q 1 M ′ ′ ′ ′ H = Q = { q , q } Then is simplicial, and so it has a base , and it can be shown 2 1 2 that every x ∈ M can be written uniquely as ′ ′ ′ ′ a ′ x = λ q + λ q + λ q + λ q + 1 1 2 2 1 1 2 2 ′ with λ ( q , a ) < 1 for all q ∈ H ∪ H 2 13 1

The idea of stratification (II) This process stops and the set of atoms of can be written as a disjoint union of H M H , … , H in such a way that 1 n M = ⟨ H ∪ ⋯ ∪ H ⟩ 1. is the basis (extreme rays, strong atoms) of H i i i n 2. for every x ∈ M there exists unique x = h + ⋯ + h n with 1 h ∈ for all M i i i for i ≥ 2 λ ( q , h + , ⋯ + h ) < 1 for all q ∈ H ∪ ⋯ ∪ H i −1 1 i n H = { H ∣ H ∣ ⋯ ∣ H } is called a stratification of H 1 2 n 14

Example x + 2 y ≡ 0 (mod 7) H = { H ∣ H ∣ H } 1 2 3 15

Example (II) x + 2 y ≡ 0 (mod 7) H = {(7, 0), (0, 7)} 1 H = {(1, 3), (5, 1)} 2 H = {(3, 2} 3 The condition λ ( q , h + ⋯ + h ) < 1 for all q ∈ H ∪ ⋯ ∪ H i −1 yields 1 i n ( x , y ) = δ (7, 0) + η (0, 7) + α (1, 3) + β (5, 1) + γ (3, 2), δ , η , α , β , γ ∈ N with , subject to α + 5 β + 3 γ < 7, 3 α + β + 2 γ < 7, γ < 2 16

General setting Let be an inside factorial monoid with set of atoms M H ˙ ˙ ˙ A stratification of is a decomposition H = H 1 ∪ 2 ∪ ⋯ ∪ k k ≥ 1 , , such that H H H for all i ∈ {1, … , k } H = , i  ∅ and the monoid M = ⟨ H ⟩ (with = ∪ ) H H ≥ i ≥ i j ≥ i i j is inside factorial with basis H i If is root-closed, then each x ∈ M ∖ {0} has a unique representation of the form M x = h + h + ⋯ + h k such that 1 2 h ∈ ⟨ H ⟩ for all i ∈ {1, … , k } i i ( h , h + ⋯ + h ) < 1 for all h ∈ H = ∪ and all i ∈ {2, … , k } λ H < i j < i M i k j 17

P. A. García-Sánchez, U. Krause, D. Llena, Inside factorial monoids and the cale monoid of a linear Diophantine equation, Journal of Algebra 531 (2019), 125–140. P. A. García-Sánchez, U. Krause, D. Llena, Strong atoms by extraction and stratified sets of atoms, preprint. Thank you for your attention 18