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

ax + by = cz,

with , and positive integers The monoid is a (full) affine monoid contained in , and it is isomorphic to , the set of nonnegative integer solutions of

ax + by ≡ 0 (mod c)

This new monoid is a (full) affine semigroup of

M a b c M N3 N N2

2

Motivation (II)

There are many procedures to find the set of atoms (minimal generators) of

N = {(x, y) ∈ N ∣

2

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 there can be different expressions (factorizations) in terms of the atoms of Elliott back in 1903 was concerned with the problem of parametrizing "uniquely" the set

• f solutions of

(x, y) A ax + by = cz

3

Inspiration

Let be a numerical semigroup (a submonoid of with finite complement in ) Let be a positive integer in ; define

Ap(S, m) = S ∖ (m + S) = {s ∈ S ∣ s − m  ∈ S}

Then every can be expressed uniquely as

x = km + w,

with and

S (N, +) N m S x ∈ S k ∈ N w ∈ Ap(S, m)

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Inspiration (II)

If is a simplicial affine Cohen-Macaulay semigroup, with a set of extreme rays and set of atoms , then every element can be expressed uniquely as

x = λ r +

1 1

⋯ + λ r +

n n

w,

with and The set has finitely many elements; these are combinations of the elements in

S Q = {r , … , r }

1 n

A x ∈ S λ ∈

i

N w ∈ Ap(S, Q) = S ∖ (Q + S) S ∖ (Q + S) A ∖ Q

5

Some definitions

Let a monoid. A submonoid

• f

is a face of if with , forces Given a subset

• f

, define the cone spanned by as

L (X) =

Q≥0

{q x +

1 1

⋯ + q x ∣

k k

k ∈ N, q ∈

i

Q , x ∈

≥0 i

X}

For an affine semigroup and an atom of , we say that is an exteme ray if is a face of An affine semigroup is simplicial if there exist linearly independent extreme rays such that An affine semigroup is Cohen-Macaulay if the semigroup ring is Cohen- Macaulay (if is simplicial it does not depend on )

M F M M x + y ∈ F x, y ∈ M x, y ∈ F X Nn X S a S a Q a

≥0

L (S)

Q≥0

S ⊆ Nn r , … , r

1 n

L (S) =

Q≥0

L ({r , … , r })

Q≥0 1 n

S K[S] S K

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Good news

The monoid

N = {(x, y) ∈ N ∣

2

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

λ(n, 0) + μ(0, m) + w

with the extreme rays of and

N (n, 0), (0, m) N w ∈ Ap(N, {(n, 0), (0, m)})

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

• f

such that for any there exists a positive integer with If is the set of atoms of , then we say that is inside factorial with base This concept generalizes that of simplicial affine semigroup We say that is root-closed if for every and a positive integer such that , we have that Every full affine semigroup is root-closed

M N M m ∈ M k km ∈ N Q N M Q M a, b ∈ M n n(a − b) ∈ M a − b ∈ M

8

Uniqueness of expressions on root-closed inside factorial monoids

Let be a root-closed inside factorial monoid with basis . Then

M = a +

a∈Ap(M,Q)

⋃ ⟨Q⟩

and this union is disjoint In particular, is a disjoint union of translates of a factorial monoid Also every element is written uniquely as a combination of elements in plus an element in

M Q M Q Ap(M, Q) = M ∖ Q + M

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Structure - Abstraction

Let be a root-closed inside factorial monoid with basis Take . Then for some and We write . With this operation becomes a torsion group, and has some special properties In fact, every root-closed inside factorial monoid is isomorphic to , with a torsion group and a free monoid, endowed with the operation

(a, f) +I (b, g) = (a + b, f + g + I(a, b))

with fulfilling the properties mentioned above

M Q a, b ∈ Ap(M, Q) a + b = c + I(a, b) c ∈ Ap(M, Q) I(a, b) ∈ ⟨Q⟩ a ⊕ b = c Ap(M, Q) I G × F G F I : G × G → F

10

Extraction grades and Apéry sets

Let be a monoid. The extraction grade for is

λ(x, y) = sup{m/n ∣ ny − mx ∈ M, m, n ∈ Z }

+

If is inside factorial with base , then

{x ∈ M ∣ λ(q, x) < 1 for all q ∈ Q} ⊆ Ap(M, Q)

Equality holds when is root-closed So every element can be written uniquely as

x = λ q +

q∈Q

∑

q

a

with for all (finite sum and )

M x, y ∈ M ∖ {0} M Q M x ∈ M λ(q, a) < 1 q ∈ Q λ ∈

q

N

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Strong atoms

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

a M Na M M Na ∩ Nb = ∅ M Q M M M

12

The idea of stratification

Assume that lives inside and that it is root-closed Trivially, is simplicial (inside factorial) and has say as extreme rays Every will we expressed uniquely as

x = λ q +

1 1

λ q +

2 2

a

with and Assume that the atoms of are . Set and Then is simplicial, and so it has a base , and it can be shown that every can be written uniquely as

x = λ q +

1 1

λ q +

2 2

λ q +

1 ′ 1 ′

λ q +

2 ′ 2 ′

a′

with for all

M N2 M Q = {q , q }

1 2

x ∈ M λ(q , a) <

1

1 λ(q , a) <

2

1 M H = Q ∪ H′ H =

1

Q M =

′

⟨H ⟩

′

M ′ H =

2

Q =

′

{q , q }

1 ′ 2 ′

x ∈ M λ(q, a ) <

′

1 q ∈ H ∪

1

H2

13

The idea of stratification (II)

This process stops and the set of atoms

• f

can be written as a disjoint union of in such a way that 1. is the basis (extreme rays, strong atoms) of

• 2. for every

there exists unique with for all for , for all is called a stratification of

H M H , … , H

1 n

Hi M =

i

⟨H ∪

i

⋯ ∪ H ⟩

n

x ∈ M x = h +

1

⋯ + hn h ∈

i

Mi i i ≥ 2 λ(q, h +

i

⋯ + h ) <

n

1 q ∈ H ∪

1

⋯ ∪ Hi−1 H = {H ∣

1

H ∣

2

⋯ ∣ H }

n

H

14

Example

x + 2y ≡ 0 (mod 7) H = {H ∣

1

H ∣

2

H }

3 15

Example (II)

The condition for all yields

(x, y) = δ(7, 0) + η(0, 7) + α(1, 3) + β(5, 1) + γ(3, 2),

with , subject to

α + 5β + 3γ < 7, 3α + β + 2γ < 7, γ < 2 x + 2y ≡ 0 (mod 7) H =

1

{(7, 0), (0, 7)} H =

2

{(1, 3), (5, 1)} H =

3

{(3, 2} λ(q, h +

i

⋯ + h ) <

n

1 q ∈ H ∪

1

⋯ ∪ Hi−1 δ, η, α, β, γ ∈ N

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General setting

Let be an inside factorial monoid with set of atoms A stratification of is a decomposition , , such that for all , and the monoid (with ) is inside factorial with basis If is root-closed, then each has a unique representation of the form such that for all for all and all

M H H H = H H ⋯ H

1∪

˙

2∪

˙ ∪ ˙

k k ≥ 1

i ∈ {1, … , k} H =

i  ∅

M =

i

⟨H ⟩

≥i

H =

≥i

∪ H

j≥i j

Hi M x ∈ M ∖ {0} x = h +

1

h +

2

⋯ + hk h ∈

i

⟨H ⟩

i

i ∈ {1, … , k} λ (h, h +

M i

⋯ + h ) <

k

1 h ∈ H =

<i

∪ H

j<i j

i ∈ {2, … , k}

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

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