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¨ oberg and Christian Gottlieb without whom I would not have done this!

Table of Contents Introduction 1 The set of patterns admitted by an ideal of a numerical semigroup 2 Non-homogeneous patterns induced by homogeneous patterns 3 Numerical semigroups as images of patterns 4 Klara Stokes

Table of Contents Introduction 1 The set of patterns admitted by an ideal of a numerical semigroup 2 Non-homogeneous patterns induced by homogeneous patterns 3 Numerical semigroups as images of patterns 4 Klara Stokes

Definition. [Bras-Amor´ os 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 a i X i such that p ( s 1 , . . . , s n ) ∈ S for all non-increasing sequences s 1 , · · · , s n ∈ S . Examples. Arf numerical semigroups are characterized by admitting the homogeneous linear “Arf pattern” X 1 + X 2 − X 3 . Homogeneous linear patterns of the form X 1 + · · · + X k − X k +1 generalise the Arf property and are called subtraction patterns [Bras-Amor´ os and Garc´ ıa-S´ anchez]. Klara Stokes

But with this definition of pattern all non-homogeneous patterns must have constant term in S . n � p (0 , . . . , 0) = a i · 0 + a 0 = a 0 ∈ S . i =1 To overcome this problem, when the non-homogeneous patterns were introduced it was with M ( S ) as domain [Bras-Amor´ os, Garc´ ıa-S´ anchez, and Vico-Oton,2013]. Numerical semigroups associated to combinatorial ( r , k )-configurations admit: ◮ X 1 + X 2 − n for n ∈ 0 , . . . , gcd( r , k ), and ◮ X 1 + · · · + X rk / 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: ◮ qX 1 − qm ( S ) if the Geil-Matsumoto bound and the Lewittes bound coincide, and ◮ ( q − 1) X 1 − ( q − 1) m ( S ) if and only if the Beelen-Ruano bound equals 1 + ( q 1) 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 Introduction 1 The set of patterns admitted by an ideal of a numerical semigroup 2 Non-homogeneous patterns induced by homogeneous patterns 3 Numerical semigroups as images of patterns 4 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 ( x 1 , . . . , x n ) and ( y 1 , . . . , y n ) are non-increasing sequences of elements in I , then so is ( x 1 + y 1 , . . . , x n + y n ). If p is a pattern admitted by I , then p ( x 1 , . . . , x n ) ∈ p ( I ) and p ( y 1 , . . . , y n ) ∈ p ( I ) and if p is linear then p ( x 1 , . . . , x n ) + p ( y 1 , . . . , y n ) = p ( x 1 + y 1 , . . . , x n + y n ) ∈ 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´ os, Garc´ ıa-S´ anchez, Vico-Oton] A linear endopattern of a numerical semigroup S is simply a linear pattern defined by a polynomial p ( X 1 , . . . , X n ) = � n i =1 a i X n + a 0 admitted by S . Therefore it has necessarily i =1 a i ≥ 0 for all n ′ ≤ n , and � n ′ constant term a 0 in S . A linear homogeneous pattern p ( X 1 , . . . , X n ) = � n i =1 a i X i is premonic if i =1 a i = 1 for some n ′ ≤ n [Bras-Amor´ � n ′ os 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 ( X 1 , . . . , X n ) = � n i =1 a i X i is a homogeneous pattern admitted by S which is not an endopattern of M ( S ), then � n i =1 a i = 0. If p ( X 1 , . . . , X n ) = � n i =1 a i X i + a 0 is a non-homogeneous pattern admitted by M ( S ) which is not an endopattern of M ( S ), then a 0 ≤ 0 and � n i =1 a i = max(0 , − a 0 / 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 ( X 1 , . . . , X n ) = � n i =1 a i X i is a homogeneous pattern admitted by S which is not an endopattern of M ( S ), then � n i =1 a i = 0. If p ( X 1 , . . . , X n ) = � n i =1 a i X i + a 0 is a non-homogeneous pattern admitted by M ( S ) which is not an endopattern of M ( S ), then a 0 ≤ 0 and � n i =1 a i = max(0 , − a 0 / 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 ( X 1 , . . . , X n ) = � n i =1 a i X i + a 0 with a 1 = 1. Proposition. Let S be a numerical semigroup. If S � = N 0 and a 0 �∈ M ( S ), then there are no monic linear patterns p ( X 1 , . . . , X n ) = � n i =1 a i X i + a 0 admitted by S or by its maximal ideal M ( S ) with � n i =1 a i = max(0 , − a 0 / m ( S )). Corollary. If S � = N 0 , 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 ( X 1 , . . . , X n ) = � n i =1 a i X i + a 0 satisfying a 0 = − ( � n i =1 a i − 1) µ ( I ) where µ ( I ) is the smallest element of I . Also, if p is a premonic endopattern of I such that a 0 = − ( � n i =1 a i − 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 ( X 1 ) = 2 X 1 is a pattern for any numerical semigroup S , but if s ∈ S then p ( s ) + s = 3 s ∈ 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