RSME Springer Series Abdallah Assi Pedro A García-Sánchez Numerical Semigroups and Applications RSME Springer Series Volume Editor-in-chief Juan Elias Garcia, Universidad de Barcelona, Barcelona, Spain Series editors Nicolas Andruskiewitsch, Universidad Nacional de Córdoba, Córdoba, Argentina María Emilia Caballero, Universidad Nacional Autónoma de México, México, Mexico Pablo Mira, Universidad Politécnica de Cartagena, Cartagena, Spain Timothy G Myers, Centre de Recerca Matemàtica, Barcelona, Spain Peregrina Quintela, Universidad de Santiago de Compostela, Santiago de Compostela, Spain Karl Schwede, University of Utah, Salt Lake City, USA As of 2015, Real Sociedad Matemática Española (RSME) and Springer cooperate in order to publish works by authors and volume editors under the auspices of a co-branded series of publications including SpringerBriefs, monographs and contributed volumes The works in the series are in English language only, aiming to offer high level research results in the fields of pure and applied mathematics to a global readership of students, researchers, professionals, and policymakers More information about this series at http://www.springer.com/series/13759 Abdallah Assi Pedro A García-Sánchez • Numerical Semigroups and Applications 123 Pedro A García-Sánchez Departamento de Álgebra Universidad de Granada Granada Spain Abdallah Assi Dèpartement de Mathématiques Université d’Angers Angers, Maine-et-Loire France ISSN 2509-8888 RSME Springer Series ISBN 978-3-319-41329-7 DOI 10.1007/978-3-319-41330-3 ISSN 2509-8896 (electronic) ISBN 978-3-319-41330-3 (eBook) Library of Congress Control Number: 2016945789 © Springer International Publishing Switzerland 2016 This work is subject to copyright All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed The use of general descriptive names, registered names, trademarks, service marks, etc in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made Printed on acid-free paper This Springer imprint is published by Springer Nature The registered company is Springer International Publishing AG Switzerland To our families Preface Early versions of this manuscript were developed for a course on numerical semigroups and their application to the study of planar curves, which was taught at the Lebanese University Since the first edition, the text has been enriched with more applications that relate numerical semigroups to ongoing research in a number of fields Nevertheless, the text is intended to be self-contained and should be accessible to beginning graduate students in mathematics We have included numerous examples and computational experiments to ensure that the reader develops a solid understanding of the fundamentals before moving forward In each case it should be possible to check the examples by hand, or by plotting the code into a computer Some of the more complicated examples can be performed with the aid of the numerical semigroups package in GAP, which is a software tool for mathematical computation available free online We begin with the basic notions and terminology relating to numerical semigroups Next, we focus on the study of irreducible numerical semigroups, and in particular free numerical semigroups, which arise in the study of planar irreducible curves Afterwards we discuss the computation of minimal presentations and how they are used to calculate nonunique factorization invariants Factorization and division are closely related, which will become apparent in studying the Feng–Rao distance and its connection to Coding Theory Numerical semigroups naturally arose as the set of values of b which have nonnegative integer solutions to Diophantine equations of the form a1 x1 ỵ ỵ an xn ¼ b, where a1,…,an,b N (here N denotes the nonnegative integers) We reduce to the case gcd(a1,…,an) = In his lectures, Frobenius asked what is the largest integer b such that a given equation has no solutions over the nonnegative integers Sylvester and others solved the n = case, and since then finding the largest such b has been known as the Frobenius problem A thorough introduction to the Frobenius problem and related topics is given in [48] An active area of study where numerical semigroups continue to play a role is within commutative algebra and algebraic geometry Let K be a field, and let A = K½ta1 ; ; tan be the K-algebra of polynomials in ta1 ; ; tan The ring A is the vii viii Preface coordinate ring of the curve parametrized by ta1 ; ; tan , and information from A can be derived from the properties of the numerical semigroup generated by the exponents a1,…,an As a result, it is often the case that names of invariants in numerical semigroup theory are inherited from Algebraic Geometry Similarly, Bertin and Carbonne [11], Delorme [23], Watanabe [58], and others have successfully identified properties of numerical semigroups which equate to their associated numerical semigroup ring fitting within various standard classifications in ring theory In the monograph [10] one can find a dictionary relating much of the overlapping terminology between commutative algebra and numerical semigroup theory Numerical semigroups are also useful in the study of singularities over planar algebraic curves Let K be an algebraically closed field of characteristic zero, and let f(x,y) be an element of K½½x; y Given another element g K½½x; y, we define the local intersection multiplicity of f with g to be the rank of the K-vector space K½½x; y=ðf ; gÞ When g runs over the set of elements of Kẵẵx; yn f ị, these numbers dene a semigroup If in addition f is irreducible, then the semigroup is a numerical semigroup This leads to a classification of irreducible formal power series in terms of their associated numerical semigroups This classification can be generalized to polynomials with one place at infinity With regard to this topic, arithmetic properties of numerical semigroups played an essential role in the proof of the Abhyankar–Moh lemma, which says that a coordinate has a unique embedding in the plane Numerical semigroups associated with planar curves are free, and thus irreducible This is why we spend some time explaining irreducible numerical semigroups and their two big subfamilies: symmetric and pseudo-symmetric numerical semigroups Recently, due to use of algebraic codes and Weierstrass numerical semigroups, some applications to coding theory and cryptography have arisen The idea is to find properties of codes in terms of an associated numerical semigroup; see for instance [21] and the references therein With this in mind we discuss Feng–Rao distances and their generalization to higher orders Another focus of recent interest has been the study of factorizations in monoids Considering the equation a1 x1 ỵ ỵ an xn ẳ b, we can think of the set of nonnegative integer solutions as the set of factorizations of b in terms of a1,…,an It can be easily shown that no numerical semigroup other than N is half-factorial, or, in other words, that there are always elements with factorizations of different lengths We will discuss some of the invariants which measure how far monoids are from being halffactorial, and how wild the sets of factorizations are Over the last decade many algorithms for computing such invariants over numerical semigroups have been developed As a result, studying these invariants over numerical semigroups offers a good chance to obtain families of examples, which can be used to test conjectures Two factorizations are expressions of the same element in terms of atoms, and one can go from one factorization to another by using a minimal presentation Hence, minimal presentations are an important tool in the study of nonunique factorization invariants We will show how to compute a minimal presentation of a Preface ix numerical semigroup both by using graphs and combinatorics and through elimination theory The graphs used to compute minimal presentations can be generalized to simplicial complexes Those having nonzero Euler characteristic are important in the expression of the generating function (Hilbert series) of the semigroup as a quotient of two polynomials The aim of this book is to give some basic notions related to numerical semigroups, and from these on the one hand to describe a classical application to the study of singularities of plane algebraic curves and on the other to show how numerical semigroups can be used to obtain handy examples of nonunique factorization invariants Angers, France Granada, Spain May 2016 Abdallah Assi Pedro A García-Sánchez Acknowledgments The first author is partially supported by the project GDR CNRS 2945 and a GENIL-SSV 2014 grant The second author is supported by the projects MTM2010-15595, MTM201455367-P, FQM-343, FQM-5849, Géanpyl (FR n 2963 du CNRS), and FEDER funds The authors would like to thank B Alarcón Heredia, M Delgado, J.I Farrán, A.M Jiménez Macas, M.J Leamer, D Llena and V Micale for their comments and corrections We would also thank the referees and the editor for their patience xi 90 Factorizations and Divisibility 5.2 Distance-Based Invariants We now introduce some invariants that depend on distances between factorizations These invariants will measure how spread is the factorizations of elements in the monoid Recall that for x = (x1 , , x p ), y = (y1 , , y p ) ∈ N p , inf{x, y} their infimum in (N p , ≤) is x ∧ y = (min{x1 , y1 }, , min{x p , y p }) The distance between x and y is defined as d(x, y) = max{|x − (x ∧ y)|, |y − (x ∧ y)|} (equivalently d(x, y) = max{|x|, |y|} − |x ∧ y|) The distance between two factorizations of the same element is lower bounded in the following way Lemma 25 Let x, y ∈ N p with x = y and ϕ(x) = ϕ(y) Then + |x| − |y| ≤ d(x, y) Proof We can assume that x ∧ y = 0, since distance is preserved under translations, ||x| − |y|| = ||x − (x ∧ y)| − |y − (x ∧ y)|| and ϕ(x − (x ∧ y)) = ϕ(y − (x ∧ y)) As ϕ(x) = ϕ(y) and x = y, in particular we have that |x| ≥ and the same for |y| Also, as x ∧ y = 0, d(x, y) = max{|x|, |y|} If |x| ≥ |y|, then + ||x| − |y|| = |x| + (2 − |y|) ≤ |x| = d(x, y) A similar argument applies for |x| ≤ |y| Example 50 The factorizations of 66 ∈ 6, 9, 11 are Z(66) = {(0, 0, 6), (1, 3, 3), (2, 6, 0), (4, 1, 3), (5, 4, 0), (8, 2, 0), (11, 0, 0)} The distance between (11, 0, 0) and (0, 0, 6) is 11 However, we can put other factorizations of 66 between them so that the maximum distance of two consecutive links (sticks in the figure) is at most 4: (11, 0, 0) (8, 2, 0) (3, 0, 0) (5, 4, 0) (0, 2, 0)|(3, 0, 0) (2, 6, 0) (0, 2, 0)|(3, 0, 0) (1, 3, 3) (0, 2, 0)|(1, 3, 0) (0, 0, 6) (0, 0, 3)|(1, 3, 0) (0, 0, 3) In the above picture the factorizations are depicted in the top of a post, and they are linked by a “catenary” labeled with the distance between two consecutive sticks On the bottom we have drawn the factorizations removing the common part with the one on the left and that of the right, respectively We will say that the catenary degree of 66 in 6, 9, 11 is at most 5.2 Distance-Based Invariants 91 A minimal presentation of S is σ = {((1, 3, 0), (0, 0, 3)), ((3, 0, 0), (0, 2, 0))} Notice also that this picture is showing us how to go from (11, 0, 0) to (0, 0, 6) by using the relations in σ For instance, as ((3, 0, 0), (0, 2, 0)) is in σ , we have that ((3 + 8, 0, 0), (0 + 8, 2, 0)) is in the congruence generated by σ , and consequently ((11, 0, 0), (8, 2, 0)) ∈ ker ϕ, that is, (11, 0, 0) and (8, 2, 0) are factorizations of the same element Since translations preserve distances, in this example, the catenary degree will be at most the maximum of d((1, 3, 0), (0, 0, 3)) = and d((3, 0, 0), (0, 0, 2)) = Given s ∈ S, x, y ∈ Z(s) and a nonnegative integer N , an N -chain joining x and y is a sequence x1 , , xk ∈ Z(s) such that • x1 = x, xk = y • For all i ∈ {1, , k − 1}, d(xi , xi+1 ) ≤ N The catenary degree of s, denoted c(s), is the least N such that for any two factorizations x, y ∈ Z(s), there is an N -chain joining them The catenary degree of S, c(S), is defined as c(S) = sup{c(s) | s ∈ S} Example 51 Let us compute the catenary degree of 77 ∈ S = 10, 11, 23, 35 We start with a complete graph with vertices the factorizations of 77 and edges labeled with the distances between them Then we remove one edge with maximum distance, and we repeat the process until we find a bridge The label of that bridge is then the catenary degree of 77 (1, 4, 1, 0) (2, 2, 0, 1) 3 (2, 2, 0, 1) 3 3 (1, 4, 1, 0) (0, 7, 0, 0) (2, 1, 2, 0) (0, 7, 0, 0) (2, 1, 2, 0) (1, 4, 1, 0) (2, 2, 0, 1) (1, 4, 1, 0) (2, 2, 0, 1) 3 3 (0, 7, 0, 0) (2, 1, 2, 0) (0, 7, 0, 0) (2, 1, 2, 0) 92 Factorizations and Divisibility Thus the catenary degree of 77 is If one looks at Proposition 48 and Example 50, one sees some interconnection between the transitivity and the way we can move from one factorization to another to minimize distances This idea is exploited in the following result Theorem 10 Let S be a numerical semigroup Then c(S) = max{c(b) | b ∈ Betti(S)} Proof Set c = maxb∈Betti(S) c(b) Clearly c ≤ c(S) Let us prove the other inequality Take s ∈ S and x, y ∈ Z(s) Let σ be a minimal presentation of S Then by Proposition 48, there exists a sequence x1 , , xk such that x1 = x, xk = y, and for every i there exists ci ∈ N p (with p the embedding dimension of S) such that (xi , xi+1 ) = (ai + ci , bi + ci ) for some (ai , bi ) such that either (ai , bi ) ∈ σ or (bi , ) ∈ σ According to Theorem 5, , bi are factorizations of a Betti element of S By using the definition of catenary degree, there is a c-chain joining and bi (also bi and ) If we add ci to all the elements of this sequence, we have a c-chain joining xi and xi+1 (distance is preserved under translations) By concatenating all these c-chains for i ∈ {1, , k − 1} we obtain a c-chain joining x and y And this proves that c(S) ≤ c, and the equality follows Example 52 With the package numericalsgps the catenary degree of an element and of the whole semigroup can be obtained as follows gap> s:=NumericalSemigroup(10,11,17,23);; gap> FactorizationsElementWRTNumericalSemigroup(60,s); [ [ 6, 0, 0, ], [ 1, 3, 1, ], [ 2, 0, 1, ] ] gap> CatenaryDegreeOfElementInNumericalSemigroup(60,s); gap> BettiElementsOfNumericalSemigroup(s); [ 33, 34, 40, 69 ] gap> List(last, x-> CatenaryDegreeOfElementInNumericalSemigroup(x,s)); [ 3, 2, 4, ] gap> CatenaryDegreeOfNumericalSemigroup(s); 5.3 How Far Is an Irreducible from Being Prime As the title suggests, the last invariant we are going to present measures how far is an irreducible from being prime Recall that a prime element is an element such that if it divides a product, then it divides one of the factors Numerical semigroups 5.3 How Far Is an Irreducible from Being Prime 93 are monoids under addition, and thus the concept of divisibility must be defined accordingly Given s, s ∈ S, recall that we write s ≤ S s if s −s ∈ S We will say that s divides s Observe that s divides s if and only if s belongs to the ideal s + S = {s +x | x ∈ S} of S If s ≤ S s , then t s divides t s in the semigroup rings K[S] and K[[S]], in the “multiplicative” sense The ω-primality of s in S, denoted ω(S, s), is the least positive integer N such that whenever s divides a1 +· · ·+an for some a1 , , an ∈ S, then s divides ai1 +· · ·+ai N for some {i , , i N } ⊆ {1, , n} Observe that an irreducible element in S (minimal generator) is prime if its ωprimality is It is easy to observe that a numerical semigroup has no primes In the above definition, we can restrict the search to sums of the form a1 +· · ·+an , with a1 , , an minimal generators of S as the following lemma shows Lemma 26 Let S be numerical semigroup and s ∈ S Then ω(S, s) is the smallest N ∈ N ∪ {∞} with the following property • For all n ∈ N and a1 , , an a sequence of minimal generators of S, if s divides a1 + · · · + an , then there exists a subset ⊂ [1, n] with cardinality less than or equal to N such that s ≤S i∈ Proof Let ω (S, s) denote the smallest integer N ∈ N0 ∪ {∞} satisfying the property mentioned in the lemma We show that ω(S, s) = ω (S, s) By definition, we have ω (S, s) ≤ ω(S, s) In order to show that ω(S, s) ≤ ω (S, s), let n ∈ N and a1 , , an ∈ S with s ≤ S a1 + · · · + an For every i ∈ [1, n] we pick a factorization = u i,1 + · · · + u i,ki with ki ∈ N and u i1 , , u i,ki minimal generators of S Then there is a subset I ∈ [1, n] and, for every i ∈ I , a subset ∅ = Λi ⊂ [1, ki ] such that #I ≤ #Λi ≤ ω (S, s) and s ≤ S i∈I and hence s ≤ S i∈I u i,ν , i∈I ν∈Λi In order to compute the ω-primality of an element s in a numerical semigroup S, one has to look at the minimal factorizations (with respect to the usual partial ordering) of the elements in the ideal s + S; this is proved in the next result Proposition 53 Let S be a numerical semigroup minimally generated by {n , , n p } Let s ∈ S Then ω(S, s) = max |m| | m ∈ Minimals≤ (Z(s + S)) 94 Factorizations and Divisibility Proof Notice that by Dickson’s lemma, the set Minimals≤ (Z(s + S)) has finitely many elements, and thus N = max |m| | m ∈ Minimals≤ (Z(s + S)) is a nonnegative integer Choose x = (x1 , , x p ) ∈ Minimals≤ (Z(s + S)) such that |x| = N Since x ∈ Z(s + S), we have that s divides s = x1 n + · · · + x p n p Assume that s divides s = y1 n + · · · + y p n p with (y1 , , y p ) < (x1 , , x p ) (that is, s divides a proper subset of summands of s ) Then s ∈ s + S, and (y1 , , y p ) ∈ Z(s + S), contradicting the minimality of x This proves that ω(S, s) ≥ N Now assume that s divides x1 n + · · · + x p n p for some x = (x1 , , x p ) ∈ N p Then x ∈ Z(s + S), and thus there exists m = (m , , m p ) ∈ Minimals≤ (Z(s + S)) with m ≤ x By definition, m n + · · · + m p n p ∈ s + S, and |m| ≤ N This proves with the help of Lemma 26 that N ≤ ω(S, s) Example 53 Let S = 3, 5, and let s = 10 ∈ S If we want to use Proposition 53, we have to compute the set Minimals≤ (Z(s + S)) To this, we find the set of nonnegative integer solutions of 3x + 5y + 7z = 10 + 3u + 5v + 7w In order to this, we can use Normaliz [13] or NormalizInterface, which is a gap interface to normaliz [35] gap> cone:=NmzCone(["inhom_equations",[[3,5,7,-3,-5,-7,-10]]]); gap> Set(NmzModuleGenerators(cone),x->x{[1 3]}); [ [ 0, 0, ], [ 0, 0, ], [ 0, 0, ], [ 0, 1, ], [ 0, 2, ], [ 1, 0, ], [ 4, 1, ], [ 5, 0, ], [ 8, 0, ] ] The minimal elements here with respect to ≤ are {(0, 0, 3), (0, 2, 0), (5, 0, 0), (1, 0, 1), (4, 1, 0), (0, 1, 2)} Hence the ω-primality of 10 in S is For numerical semigroups these computations can be performed using Apéry sets (see [12, Remarks 5.9]) gap> s:=NumericalSemigroup(3,5,7);; gap> OmegaPrimalityOfElementInNumericalSemigroup(10,s); For S a numerical semigroup minimally generated by {n , , n p }, the ωprimality of S is defines as ω(S) = max{ω(S, n i ) | i ∈ {1, , p}} 5.3 How Far Is an Irreducible from Being Prime 95 We are going to relate Delta sets with catenary degree and ω-primality To this end we need the following technical lemma For b = (b1 , , b p ) ∈ N p , define Supp(b) = {i ∈ {1, , p} | bi = 0} Lemma 27 Let S be a numerical semigroups minimally generated by {n , , n p }, and let n ∈ Betti(S) Let a, b ∈ Z(n) in different R-classes For every i ∈ Supp(b) we have that a ∈ Minimals≤ Z(n i + S) Proof Assume to the contrary that there exists c ∈ Z(n i + S) and x ∈ Nk \ {0} such that c + x = a From c < a, a · b = and i ∈ Supp(b), we deduce that i ∈ / Supp(c) As c ∈ Z(n i + S), there exists d ∈ Z(n i + S) with i ∈ Supp(d) and ϕ(c) = ϕ(d) Hence ϕ(d + x) = ϕ(c + x) = ϕ(a) Moreover (d + x) · (c + x) = (d + x) · a = 0, and (d + x) · b = 0, which leads to aRb, a contradiction Theorem 11 Let S be a numerical semigroup Then max (S) + ≤ c(S) ≤ ω(S) Proof Assume that d ∈ (S) Then there exists s ∈ S and x, y ∈ Z(s) such that |x| < |y|, d = |y| − |x| and there is no z ∈ Z(s) with |x| < |z| < |y| From the definition of c(S), there is a c(S)-chain z , , z k joining x and y As in the proof of Theorem 9, we deduce that there exists i such that |z i | < |x| < |y| < |z i+1 | Then + d = + |y| − |x| ≤ + |z i+2 | − |z i |, and by Lemma 25, + |z i+2 | − |z i | ≤ d(z i , z i+1 ) The definition of c(S)-chain implies that d(z i , z i+1 ) ≤ c(S) Hence + d ≤ c(S), and consequently max (S) + ≤ c(S) Let σ be a minimal presentation of S For every (a, b) ∈ σ , there exists n i and n j minimal generators such that a ∈ Minimals≤ Z(n i + S) and b ∈ Minimals≤ Z(n j + S) (Lemma 27) From the definition of ω(S), both |a| and |b| are smaller than or equal to ω(S) Set c = max{max{|a|, |b|} | (a, b) ∈ σ } Then c ≤ ω(S) Now we prove that c(S) ≤ c Let s ∈ S and x, y ∈ Z(s) Then ϕ(x) = ϕ(y) and as σ is a presentation, by Proposition 48, there exists a sequence x1 , , xk ∈ N p ( p = e(S)) such that x1 = x, xk = y and for every i there exists , bi , ci ∈ N p such that (xi , xi+1 ) = (ai + ci , bi + ci ), with either (ai , bi ) ∈ σ or (bi , ) ∈ σ Notice that d(xi , xi+1 ) = d(ai , bi ) = max{|ai |, |bi |} (ai and bi are in different R-classes and thus ·bi = 0, or equivalently, ∧bi = 0) Hence d(xi , xi+1 ) ≤ c, and consequently x1 , , xk is a c-chain joining x and y This implies that c(S) ≤ c, and we are done Example 54 Let us go back to S = 10, 11, 17, 23 From Example 49 and Theorem 9, we know that max (S) = gap> OmegaPrimalityOfNumericalSemigroup(s); From Theorem 11, we deduce that c(S) ∈ {5, 6} Recall that by Example 52, we know that c(S) = 96 Factorizations and Divisibility There are many other nonunique factorization invariants that can be defined on any numerical semigroup It was our intention just to show some of them and the last theorem that relates these invariants coming from lengths, distances and primality (respectively), and at the same time show how minimal presentations can be used to study them The reader interested in this topic is referred to [33], and for a review of the computational aspects to [32] 5.4 Divisors and Feng–Rao Distances We have used several times the partial order ≤ S with S a numerical semigroup In this section we introduce a concept that has been widely used in one point algebraic geometry codes associated to curves Let s ∈ S Define the set of divisors of s as D(s) = {n ∈ S | s − n ∈ S} Lemma 28 D(s) = S ∩ (s − S) Proof An integer n ∈ D(S) if and only if n ∈ S and s − n = t for some t ∈ S, and this is equivalent to n ∈ S ∩ (s − S) Let q be a power of a prime number, and let Fq be the finite field with q elements Let R be the affine coordinate ring of an absolutely irreducible nonsingular curve over Fq with a single rational point at infinity, say Q Let P = {P1 , , Pn } be a set of n distinct affine Fq -rational points of the curve Define the evaluation map on P as follows (see [14]) evP : R → Fqn , evP ( f ) = ( f (P1 ), , f (Pn )) Set L(m Q) = { f ∈ R | v Q ( f ) ≥ −m}, where v Q is the discrete valuation at Q The image of −v Q of R ∗ is a numerical semigroup, say S The function −v Q applied to f ∈ R is measuring the order of the pole Q in f Since f is in R, if Q is a pole, then it is the only pole of f Notice also that if f is a meromorphic series whose only pole is Q (the infinity), then f is forced to be a polynomial Hence S is the Weierstrass semigroup of the curve at Q Observe that L(0Q) ⊆ L(1Q) ⊆ L(2Q) ⊆ · · · and that equalities in this chain correspond to the gaps in the Weierstrass semigroup For s ∈ S, denote by Cs the orthogonal (with respect to the usual dot product) of evP (L(s Q)), which is called the one point algebraic code defined by s Q and P It is well known that the minimum distance of Cs it as least the Feng–Rao distance (or order bound; called after Feng and Rao since it was first introduced in [27]) of s, which is defined as 5.4 Divisors and Feng–Rao Distances δ F R (s) = 97 D(s ) | s ≤ s and s ∈ S The asymptotic behavior of the Feng–Rao distance is easy to compute thanks to the following result (see [20, Lemma 5]) Lemma 29 Let c and g be the conductor and genus of S, respectively Let s be an integer with s ≥ 2c − Then D(s) = s + − 2g Proof Lemma 28 asserts that D(s) = (S∩(s−S)) Observe that S∩(s−S) ⊆ [0, s], and so at most we have s + divisors of s Also s ≥ 2c − 1, which means that it is larger than the Frobenius number of S So the gaps of S are not in D(s), and there are exactly g of them in [0, c] If h is a gap of S, then s − h ∈ / s − S, and thus we are counting g more elements in [s − c, s] that are not in s − S Any other integer in [0, s] is both in S and in s − S And this completes the proof The number s + − 2g is sometimes known as the Goppa bound Proposition 54 Let S be a numerical semigroup and let s ≥ 2c − 1, with c the conductor of S Then δ F R (s) = s + − 2g Proof By definition and Lemma 29, δ F R (s) = min{s + − 2g | s ≤ s and s ∈ S}, which clearly equals s + − 2g GAP example 12 Let us see this behavior with S = 3, 5, gap> ndiv:=function(x,s) > return Length(Filtered(Intersection([0 x],s), > y->x-y in s)); > end; function( x, s ) end gap> s:=NumericalSemigroup(3,5,7);; gap> c:=ConductorOfNumericalSemigroup(s); gap> List(Intersection([0 15],s), x->ndiv(x,s)); [ 1, 2, 2, 3, 2, 4, 4, 5, 6, 7, 8, 9, 10 ] The advantage of the Feng–Rao distance is that is computed directly on the semigroup There is a generalization of the Feng–Rao distance, which is known as generalized Feng–Rao distance: δrF R : S −→ N, s → D(s1 , , sr ) s ≤ s1 ≤ · · · ≤ sr , s1 , , sr ∈ S where D(s1 , , sr ) = ri=1 D(si ) As expected its computation is harder than the classical Feng–Rao distance (see for instance [21] for the calculation of this function 98 Factorizations and Divisibility on numerical semigroups with embedding dimension two or [20] for semigroups generated by intervals) For r = 2, we will see next how the asymptotic behavior of this new distance is related with the Apéry sets that we have used many times in this manuscript Recall that we defined the Apéry set of s ∈ S\{0} as the set of elements n in S such that n − s ∈ / S We can extend this definition to any positive integer x, though we already saw that the resulting set will no longer have x elements (GAP example 1) So let x ∈ N\{0} Define the Apéry set of x in S as Ap(S, x) = {s ∈ S | s − x ∈ / S} Selmer’s formula for the Frobenius number still holds (Proposition 5) Lemma 30 Let S be a numerical semigroup and let n be a positive integer Then F(S) + n = max(Ap(S, n)) Proof Let x > F(S) + n Then by definition of Frobenius number x ∈ S, and / Ap(S, n) Clearly, F(S) + n ∈ Ap(S, n) x − n > F(S), which implies that x ∈ A desert of S is a maximal interval of gaps of S (counting also the set of all nonnegative integers) The following curiosity is a straight consequence of the definition Proposition 55 The number of deserts of S is Ap(S, 1) In order to compute the generalized Feng–Rao distances of order two, we make use of the following characterization [21, Proposition 11] Lemma 31 Let m ≥ 2c −1, with c the conductor of S, and let n be a positive integer Then D(m + n) \ D(m) = m + n − Ap(S, n) / S This implies Proof Take s ∈ D(m + n) \ D(m) Then m + n − s ∈ S and m − s ∈ that m + n − s ∈ Ap(S, n), whence s ∈ m + n − Ap(S, n) Now take s ∈ m + n − Ap(S, n), say s = m + n − w with w ∈ Ap(S, n) Notice that m +n −(m +n −w) = w ∈ S, and m −(m +n −w) = w −n ∈ / S So it remains to prove that m + n − w ∈ S In light of Lemma 30, w ≤ F(S) + n, and consequently n − w ≥ −F(S) Thus m + n − w ≥ 2c − − F(S) = 2c − − (c − 1) = c, which yields m + n − w ∈ S Hence (D(m + n) \ D(m)) = Ap(S, n) This implies that for every pair of integers m and n such that m ≥ 2c − and n > 0, we have that D(m, m + n) = m + − 2g + Ap(S, n) With this we get the following result describing the asymptotic behavior of δ 2F R Proposition 56 Let S be a numerical semigroup with conductor c Then for all m ≥ 2c − 1, δ 2F R (m) = m + − 2g + Ap(S, n) | n ∈ N∗ 5.4 Divisors and Feng–Rao Distances 99 Observe that the amount { Ap(S, n) | n ∈ N∗ } is fixed, that is, δ 2F R (m) is a translation of m for m big enough Example 55 Let S = 3, We know that for n ∈ S, Ap(S, n) = n So in order to calculate the minimum of all Ap(S, n) with n a positive integer, it suffices to calculate this 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(1973) 61 A Zhai, Fibonacci-like growth of numerical semigroups of a given genus Semigroup Forum 86, 634–662 (2013) Index A Apéry set, Apéry set of an integer, 98 Approximate root, 46 B Betti element, 75 C Catenary degree, 91 Chain of factorizations, 91 Characteristic sequences, 36 Conductor of a numerical semigroup, Congruence compatible, 73 generated by a set, 70 Contact, 48 Curve with one place at infinity, 58 D Degree of singularity, Delta set, 87 Desert, 98 Distance factorizations, 90 Divide, 93 E Elasticity, 86 Embedding dimension of a numerical semigroup, Expansion, 40 F Factorization homomorphism, 70 Feng–Rao distance, 96 generalized, 97 Frobenius number, G G-adic expansion, 40 Gap of a numerical semigroup, Generating function, 80 Genus of a numerical semigroup, Graph associated to an element, 73 H Half-factorial monoid, 85 Hilbert series, 80 I Intersection multiplicity, 37 L Length of a factorization, 85 M Milnor number, 57 Minimal set of generators, Multiplicity of a numerical semigroup, N Newton–Puisieux exponents, 36 Numerical semigroup, © Springer International Publishing Switzerland 2016 A Assi and P.A García-Sánchez, Numerical Semigroups and Applications, RSME Springer Series, DOI 10.1007/978-3-319-41330-3 105 106 associated to a curve, 61 associated to a polynomial, 55 free, 26 irreducible, 17 maximal embedding dimension, 11 pseudo-symmetric, 19 symmetric, 18 telescopic, 26 O ω-primality, 93 One point algebraic code, 96 Order, 31 Oversemigroup of a numerical semigroup, 15 P Presentation of a numerical semigroup, 71 Pseudo-approximate root, 37 Pseudo-Frobenius numbers, Index S Semigroup ring, Set of divisors, 96 Set of factorizations, 72 Set of lengths, 85 Shaded set, 80 Special gaps of a numerical semigroup, 14 Standard representation in a free numerical semigroup, 27 Support, 34 T Tschirnhausen transform, 44 Type of a numerical semigroup, W Weirstrass semigroup, ... Publishing Switzerland 2016 A Assi and P.A García-Sánchez, Numerical Semigroups and Applications, RSME Springer Series, DOI 10.1007/978-3-319-41330-3_2 17 18 Irreducible Numerical Semigroups 2.1... to numerical semigroups, and from these on the one hand to describe a classical application to the study of singularities of plane algebraic curves and on the other to show how numerical semigroups. .. usually denoted K[S] and is called the semigroup ring of S over K © Springer International Publishing Switzerland 2016 A Assi and P.A García-Sánchez, Numerical Semigroups and Applications, RSME