Patterns admitted by a numerical semigroup Klara Stokes With - - PowerPoint PPT Presentation

Patterns admitted by a numerical semigroup

Klara Stokes With gratitude to Ralf Fr¨

• berg and Christian Gottlieb

without whom I would not have done this!

Table of Contents

1

Introduction

2

The set of patterns admitted by an ideal of a numerical semigroup

3

Non-homogeneous patterns induced by homogeneous patterns

4

Numerical semigroups as images of patterns

Klara Stokes

Table of Contents

1

Introduction

2

The set of patterns admitted by an ideal of a numerical semigroup

3

Non-homogeneous patterns induced by homogeneous patterns

4

Numerical semigroups as images of patterns

Klara Stokes

• Definition. [Bras-Amor´
• s and Garc´

ıa-S´ anchez, 2006] A homogeneous pattern admitted by a numerical semigroup S is a homogeneous linear multivariate polynomial p = n

i=1 aiXi such that

p(s1, . . . , sn) ∈ S for all non-increasing sequences s1, · · · , sn ∈ S. Examples. Arf numerical semigroups are characterized by admitting the homogeneous linear “Arf pattern” X1 + X2 − X3. Homogeneous linear patterns of the form X1 + · · · + Xk − Xk+1 generalise the Arf property and are called subtraction patterns [Bras-Amor´

• s and Garc´

ıa-S´ anchez].

Klara Stokes

But with this definition of pattern all non-homogeneous patterns must have constant term in S. p(0, . . . , 0) =

n

• i=1

ai · 0 + a0 = a0 ∈ S. To overcome this problem, when the non-homogeneous patterns were introduced it was with M(S) as domain [Bras-Amor´

• s, Garc´

ıa-S´ anchez, and Vico-Oton,2013]. Numerical semigroups associated to combinatorial (r, k)-configurations admit:

◮ X1 + X2 − n for n ∈ 0, . . . , gcd(r, k), and ◮ X1 + · · · + Xrk/ gcd(r,k) + 1.

Weierstrass semigroups S of multiplicity m(S) of a rational place of a function field over a finite field of cardinality q admit:

◮ qX1 − qm(S) if the Geil-Matsumoto bound and the Lewittes bound

coincide, and

◮ (q − 1)X1 − (q − 1)m(S) if and only if the Beelen-Ruano bound equals

1 + (q1)m.

Klara Stokes

But now we have two different definitions of patterns. Let’s generalise and unify! First, remember: A relative ideal of a numerical semigroup S is a set H ⊆ Z satisfying H + S ⊆ H and H + d ⊆ S for some d ∈ S. A relative ideal contained in S is an ideal of S. An ideal is proper if it is distinct from S. The set of proper ideals of S has a maximal element with respect to inclusion. This ideal is called the maximal ideal of S, and equals M(S), the set of non-zero elements of S. The dual of a relative ideal H is the relative ideal H∗ = (S − H) := {x ∈ Z : x + H ⊆ S}.

Klara Stokes

Definition. A pattern admitted by an ideal I of a numerical semigroup S is a multivariate polynomial function which returns an element in S when evaluated on any non-increasing sequence of elements from I. We say that the ideal I admits the pattern.

Klara Stokes

Definition. A pattern admitted by an ideal I of a numerical semigroup S is a multivariate polynomial function which returns an element in S when evaluated on any non-increasing sequence of elements from I. We say that the ideal I admits the pattern. If I = S, then we say that the numerical semigroup S admits the pattern. What happened with the previous definitions of patterns? (Homogeneous) patterns evaluted on S have become patterns admitted by S. (Non-homogeneous) patterns evaluated on M(S) have become patterns admitted by M(S). Note that a pattern admitted by an ideal I of a numerical semigroup S is also admitted by any ideal J ⊆ I.

Klara Stokes

We identify the pattern with its polynomial. We say that the pattern is linear and homogeneous, when the pattern polynomial is linear and homogeneous. The length of a pattern: the number of indeterminates. The degree of a pattern: the degree of the pattern polynomial. One pattern p induces another pattern q if any ideal of a numerical semigroup that admits p also admits q. Two patterns are equivalent if they induce each other.

Klara Stokes

Table of Contents

1

Introduction

2

The set of patterns admitted by an ideal of a numerical semigroup

3

Non-homogeneous patterns induced by homogeneous patterns

4

Numerical semigroups as images of patterns

Klara Stokes

Lemma. Let p be a pattern admitted by an ideal I of a numerical semigroup. If p is linear then p(I) is closed under addition.

Klara Stokes

Lemma. Let p be a pattern admitted by an ideal I of a numerical semigroup. If p is linear then p(I) is closed under addition. Proof. If (x1, . . . , xn) and (y1, . . . , yn) are non-increasing sequences of elements in I, then so is (x1 + y1, . . . , xn + yn). If p is a pattern admitted by I, then p(x1, . . . , xn) ∈ p(I) and p(y1, . . . , yn) ∈ p(I) and if p is linear then p(x1, . . . , xn) + p(y1, . . . , yn) = p(x1 + y1, . . . , xn + yn) ∈ p(I).

Klara Stokes

Definition. Let p be a linear pattern admitted by an ideal I of a numerical semigroup. If p(I) ⊆ I, then we say that p is an endopattern of I. Definition. If additionally p is surjective, that is, if p(I) = I, then we say that p is a surjective endopattern of I.

Klara Stokes

• Lemma. [Bras-Amor´
• s, Garc´

ıa-S´ anchez, Vico-Oton] A linear endopattern of a numerical semigroup S is simply a linear pattern defined by a polynomial p(X1, . . . , Xn) = n

i=1 aiXn + a0 admitted by S.

Therefore it has necessarily n′

i=1 ai ≥ 0 for all n′ ≤ n, and

constant term a0 in S. A linear homogeneous pattern p(X1, . . . , Xn) = n

i=1 aiXi is premonic if

n′

i=1 ai = 1 for some n′ ≤ n [Bras-Amor´

• s and Garc´

ıa-S´ anchez]. Lemma. Any linear surjective endopattern of a numerical semigroup S is necessarily a linear homogeneous patterns admitted by S. If p is a premonic homogeneous endopattern of S, then p is always surjective.

Klara Stokes

Lemma. Let S be a numerical semigroup and M(S) its maximal ideal. If p(X1, . . . , Xn) = n

i=1 aiXi is a homogeneous pattern admitted by

S which is not an endopattern of M(S), then n

i=1 ai = 0.

If p(X1, . . . , Xn) = n

i=1 aiXi + a0 is a non-homogeneous pattern

admitted by M(S) which is not an endopattern of M(S), then a0 ≤ 0 and n

i=1 ai = max(0, −a0/m(S)).

So many important homogeneous and non-homogeneous patterns are endopatterns of M(S)!

Klara Stokes

Lemma. Let S be a numerical semigroup and M(S) its maximal ideal. If p(X1, . . . , Xn) = n

i=1 aiXi is a homogeneous pattern admitted by

S which is not an endopattern of M(S), then n

i=1 ai = 0.

If p(X1, . . . , Xn) = n

i=1 aiXi + a0 is a non-homogeneous pattern

admitted by M(S) which is not an endopattern of M(S), then a0 ≤ 0 and n

i=1 ai = max(0, −a0/m(S)).

So many important homogeneous and non-homogeneous patterns are endopatterns of M(S)! Example. If p is a pattern of S then p is also an endopattern of M(S) if p is the Arf pattern, a subtraction patterns, or a pattern X + a with a pseudo-Frobenius of S. They all belong to the important class of monic linear patterns.

Klara Stokes

A monic linear pattern is defined by a linear polynomial p(X1, . . . , Xn) = n

i=1 aiXi + a0 with a1 = 1.

Proposition. Let S be a numerical semigroup. If S = N0 and a0 ∈ M(S), then there are no monic linear patterns p(X1, . . . , Xn) = n

i=1 aiXi + a0 admitted by S or

by its maximal ideal M(S) with n

i=1 ai = max(0, −a0/m(S)).

Corollary. If S = N0, then any monic linear pattern admitted by M(S) is an endopattern of M(S). Lemma. Any linear surjective endopattern of a proper ideal I of a semigroup S is necessarily of the form p(X1, . . . , Xn) = n

i=1 aiXi + a0 satisfying

a0 = −(n

i=1 ai − 1)µ(I) where µ(I) is the smallest element of I. Also, if

p is a premonic endopattern of I such that a0 = −(n

i=1 ai − 1)µ(I),

then p is surjective.

Klara Stokes

We have seen that if p is a premonic linear homogeneous pattern admitted by a numerical semigroup S, then p(S) = S. In particular, the image of a monic linear homogeneous pattern admitted by S equals S. More generally: Lemma. Let p be a monic linear pattern admitted by the numerical semigroup S or by its maximal ideal M(S). Then p(S), or p(M(S)), respectively, is an ideal of S.

Klara Stokes

In general it is not true that the image of a linear pattern is an ideal of a numerical semigroup. Example. The pattern p(X1) = 2X1 is a pattern for any numerical semigroup S, but if s ∈ S then p(s) + s = 3s ∈ p(S) implies that 2|s which cannot be true for all elements in S. Therefore p(S) + S p(S) and p(S) is not an ideal of S.

Klara Stokes

Lemma. Let I be an ideal of a numerical semigroup S and suppose that p and q are two patterns admitted by I. Then p + q and rp are also patterns admitted by I for any polynomial r with coefficients in Z such that r(I) ≥ 0 when evaluated on any non-increasing sequence of elements from I.

Klara Stokes

Lemma. Let I be an ideal of a numerical semigroup S and suppose that p and q are two patterns admitted by I. Then p + q and rp are also patterns admitted by I for any polynomial r with coefficients in Z such that r(I) ≥ 0 when evaluated on any non-increasing sequence of elements from I. Lemma. If additionally p and q are endopatterns of I and r(I) > 0 when evaluated

• n any non-increasing sequence of elements from I, then

p + q and rp are also endopatterns of I.

Klara Stokes

Denote by Pd

n (I) and Ed n (I) the set of patterns and endopatterns

respectively, of length at most n and degree at most d, admitted by the ideal I of a numerical semigroup S. Then P1

n(I) is the set of linear patterns of length at most n admitted by I.

Proposition. Let I be an ideal of a numerical semigroup S. Then Pd

n (I) is a semigroup with zero, a monoid, and

if I = S, then Ed

n (I) is a semigroup without zero.

Klara Stokes

The set Pd

n (I) is not preserved by polynomial multiplication, but, if so is

desired, this problem can be overcome by instead considering patterns of arbitrary degree. Denote by Pn(I) the set of patterns of length at most n that are admitted by I. Proposition. Let I be an ideal of a numerical semigroup S. Then Pn(I) is a semiring. There is a unit element if and only if I = N0. Proposition. Let I be an ideal of a numerical semigroup S and consider the set of polynomials R(I) = {r ∈ Z[X1, . . . , Xn] : r(s1, . . . , sn) ≥ 0 ∀s1 ≥ · · · ≥ sn ∈ I}. Then R(I) is a semiring (with zero and unit elements) and Pn(I) ∪ {0} is an R(I)-(semigroup)algebra.

Klara Stokes

Table of Contents

1

Introduction

2

The set of patterns admitted by an ideal of a numerical semigroup

3

Non-homogeneous patterns induced by homogeneous patterns

4

Numerical semigroups as images of patterns

Klara Stokes

Let S be a numerical semigroup and let p(X1, . . . , Xn) = n

i=1 aiXi + a0, and

q(X1, . . . , Xn) = p − a0. When is p induced by q? Lemma. If a0 ∈ S ∪ PF(S), then the non-homogeneous pattern defined by p is induced by the homogeneous pattern defined by q. Proof. If q(s1, . . . , sn) ∈ S and a0 ∈ S ∪ PF(S), then q(s1, . . . , sn) + a0 ∈ S.

Klara Stokes

Let S be a numerical semigroup and let p(X1, . . . , Xn) = n

i=1 aiXi + a0, and

q(X1, . . . , Xn) = p − a0. When is p induced by q? Lemma. If a0 ∈ S ∪ PF(S), then the non-homogeneous pattern defined by p is induced by the homogeneous pattern defined by q. Proof. If q(s1, . . . , sn) ∈ S and a0 ∈ S ∪ PF(S), then q(s1, . . . , sn) + a0 ∈ S. Example. Let S = 2, 7 = {0, 2, 4, 6, →}, then PF(S) = {5}. S admits the Arf pattern X1 + X2 − X3 and the non-homogeneous pattern X1 + X2 − X3 + 5, but not X1 + X2 − X3 + 3. However, 1 ∈ S ∪ PF(S) but X1 + X2 + X3 + 1 is a pattern of S.

Klara Stokes

The polynomial X1 + a0 is a pattern admitted by S if and only if a0 ∈ S ∪ PF(S). By replacing the variable X1 by a pattern q admitted by S this resulted in a statement on for which a0 ∈ Z the pattern q induces the pattern q + a0. Now generalise this idea to sums of several patterns! Definition. For d ≥ 1, define the set PF d(S) = (S − dM) \ (S − (d − 1)M) and call it the set of elements at distance d from S. The elements at distance one are PF 1(S) = PF(S), the set of pseudo-Frobenius of S. The elements at distance zero from S can be defined to be the elements in S.

Klara Stokes

This is not the only way to generalise the notion of pseudo-Frobenius! Let S = {0 = s0, s1, . . . , sn, →} be a numerical semigroup with conductor sn. For 1 ≤ i ≤ n, Si = {s ∈ S : s ≥ si} is an ideal of S and S(i) = S∗

i = (S − Si) is its dual relative ideal.

Define Ti(S) = S(i) \ S(i − 1). The type sequence of a numerical semigroup S is the finite sequence (|Ti(S)| : 1 ≤ i ≤ n) [Barucci,Dobbs and Fontana,1997]. Note that T1 = PF and |PF| is the type of S, so this is a generalisation of pseudo-Frobenius.

Klara Stokes

Let S be a numerical semigroup and M = M(S) its maximal ideal. The Lipman semigroup of S is L(S) =

h≥1(hM − hM) [Lipman, 1971].

Klara Stokes

Proposition. The cardinality of PF d(S) converges to m = m(S). The convergence follows the convergence of the Lipman semigroup of S. Proof. Consider the Lipman semigroup L(S) =

h≥1(hM − hM).

There is a smallest h0 ≥ 1 such that L(S) = hM − hM whenever h ≥ h0, in which case we also have (h + 1)M = hM + m. [Barucci, Dobbs and Fontana, 1997]) Therefore, if d ≥ h0 then z + (d + 1)M = z + m + dM for z ∈ Z implying that z + (d + 1)M ⊆ S and z + dM ⊆ S if and only if (z + m) + dM ⊆ S and (z + m) + (d − 1)M ⊆ S. Consequently (S − (d + 1)M) = (S − dM) − m so that PF d+1(S) = (S − (d + 1)M) \ (S − dM) = ((S − dM) − m) \ (S − dM), which has cardinality m.

• Klara Stokes

Remember that P1

1(S) denotes the semigroup of linear patterns in one

variable. Can we determine P1

1(S) in general?

Proposition. For any numerical semigroup S we have P1

1(S) ⊇ {p(X1) = a1X1 + a0 ∈ Z[X1] : a1 ≥ 0, a0 ∈ S ∪ a1 i=1 PF i(S)}.

A numerical semigroup S is ordinary if z ∈ S for all z ∈ Z such that z ≥ m(S). Proposition. If S is an ordinary numerical semigroup, then P1

1(S) = {p(X1) = a1X1 + a0 ∈ Z[X1] : a1 ≥ 0, a0 ∈ S ∪ a1 i=1 PF i(S)}.

Klara Stokes

Table of Contents

1

Introduction

2

The set of patterns admitted by an ideal of a numerical semigroup

3

Non-homogeneous patterns induced by homogeneous patterns

4

Numerical semigroups as images of patterns

Klara Stokes

Lemma. Let p(X1, . . . , Xn) = a1X1 + · · · + anXn be a homogeneous linear pattern admitted by the numerical semigroup S. Then p(S) is a numerical semigroup if and only if gcd(a1, . . . , an) = 1. Corollary. Let p(X1, . . . , Xn) = a1X1 + · · · + anXn be a homogeneous linear endopattern admitted by the maximal ideal M(S) of a numerical semigroup S. Then p(M(S)) is the maximal ideal of a numerical semigroup if and only if gcd(a1, . . . , an) = 1. Note: if p(M(S)) ⊆ M(S) it could happen that 0 ∈ p(M(S)), which would imply that p(M(S)) is a numerical semigroup and not a maximal ideal.

Klara Stokes

What numerical semigroups can be obtained as the image of another numerical semigroup under a pattern? Proposition. Any numerical semigroup S = a1, . . . , ae is the image of N0 under the homogeneous pattern p(X1, . . . , Xe) = a1X1 + e

i=2(ai − ai−1)Xi.

Klara Stokes

What numerical semigroups can be obtained as the image of another numerical semigroup under a pattern? Proposition. Any numerical semigroup S = a1, . . . , ae is the image of N0 under the homogeneous pattern p(X1, . . . , Xe) = a1X1 + e

i=2(ai − ai−1)Xi.

Note that if the numerical semigroup S is the image of a numerical semigroup S′ ⊇ S under a pattern p, then S admits p. Therefore we can consider the chain of numerical semigroups N0 ⊇ p(N0) ⊇ p(p(N0)) ⊇ · · · .

Klara Stokes

Thank you for listening!

Klara Stokes