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In this paper, we study a family of binomial ideals defining monomial curves in the n-dimensional affine space determined by n hypersurfaces of the form $x_i^{c_i} - x_1^{u_{i1}} \cdots x_n^{u_{1n}}$ in $\Bbbk [x_1, \ldots , x_n]$ with $u_{ii} = 0, \ i\in \{ 1, \ldots , n\}$ . We prove that the monomial curves in that family are set-theoretic complete intersections. Moreover, if the monomial curve is irreducible, we compute some invariants such as genus, type and Frobenius number of the corresponding numerical semigroup. We also describe a method to produce set-theoretic complete intersection semigroup ideals of arbitrary large height.

We give an algorithm to determine whether Wilf’s conjecture holds for all numerical semigroups with a given multiplicity [Formula: see text], and use it to prove Wilf’s conjecture holds whenever [Formula: see text]. Our algorithm utilizes techniques from polyhedral geometry, and includes a parallelizable algorithm for enumerating the faces of any polyhedral cone up to orbits of an automorphism group. We also introduce a new method of verifying Wilf’s conjecture via a combinatorially flavored game played on the elements of a certain finite poset.

In this manuscript we show that the second Feng–Rao number of any telescopic numerical semigroup agrees with the multiplicity of the semigroup. To achieve this result we first study the behavior of Apery sets under gluings of numerical semigroups. These results provide a bound for the second Hamming weight of one-point Algebraic Geometry codes, which improves upon other estimates such as the Griesmer Order Bound.

This work extends the results known for the Delta sets of non-symmetric numerical semigroups with embedding dimension three to the symmetric case. Thus, we have a fast algorithm to compute the Delta set of any embedding dimension three numerical semigroup. Also, as a consequence of these resutls, the sets that can be realized as Delta sets of numerical semigroups of embedding dimension three are fully characterized.

Let f 1 , ź , f s be formal power series (respectively polynomials) in the variable x. We study the semigroup of orders of the formal series in the algebra K ź f 1 , ź , f s ź ź K ź x ź (respectively the semigroup of degrees of polynomials in K f 1 , ź , f s ź K x ). We give procedures to compute these semigroups and several applications. We prove in particular that the space curve parametrized by f 1 , ź , f s has a flat deformation into a monomial curve.

This work extends the results known for the Delta sets of non-symmetric numerical semigroups with embedding dimension three to the symmetric case. Thus, we have a fast algorithm to compute the Delta set of any embedding dimension three numerical semigroup. Also, as a consequence of these results, the sets that can be realized as Delta sets of numerical semigroups of embedding dimension three are fully characterized.

Producción Científica, We describe the second (generalized) Feng-Rao distance for elements in an Arf numerical semigroup that are greater than or equal to the conductor of the semigroup. This provides a lower bound for the second Hamming weight for one point AG codes. In particular, we can obtain the second Feng-Rao distance for the codes defined by asymptotically good towers of function fields whose Weierstrass semigroups are inductive. In addition, we compute the second Feng-Rao number, and provide some examples and comparisons with previous results on this topic. These calculations rely on Apéry sets, and thus several results concerning Apéry sets of Arf semigroups are presented., Ministerio de Economía, Industria y Competitividad; y Fondo Europeo de Desarrollo Regional FEDER( Projects MTM2014-55367-P / MTM2015-65764-C3-1-P), Junta de Andalucía (Grant FQM-343), Fundação para a Ciência e a Tecnologia (Project UID/MAT/00297/2013)

The second Feng-Rao number of every inductive numerical semigroup is explicitly computed. This number determines the asymptotical behavior of the order bound for the second Hamming weight of one-point algebraic geometry codes. In particular, this result is applied for the codes defined by asymptotically good towers of function fields whose Weierstrass semigroups are inductive. In addition, some properties of inductive numerical semigroups are studied, the involved Apery sets are computed in a recursive way, and some tests to check whether given numerical semigroups are inductive or not are provided.

We characterize numerical semigroups of embedding dimension three having the same catenary and tame degrees., García Sánchez is supported by the projects MTM2010-15595, FQM-343, FQM-5849, and FEDER funds. The contents of this article are part of Viola’s master’s thesis. Part of this work was done while she visited the Univerisidad de Granada under the European Erasmus mobility program.

Every almost symmetric numerical semigroup can be constructed by removing some minimal generators from an irreducible numerical semigroup with its same Frobenius number.

We give a positive answer to the Huneke–Wiegand Conjecture for monomial ideals over free numerical semigroup rings, and for two-generated monomial ideals over complete intersection numerical semigroup rings.

Let \({\mathbb K}\) be an algebraically closed field of characteristic zero and let \(f(x,y)=y^n+a_1(x)y^{n-1}+\dots +a_n(x)\) be a nonzero polynomial of \({\mathbb K}(\!(x)\!)[y]\) where \({\mathbb K}(\!(x)\!)\) denotes the field of meromorphic series in x.

Electronic version of an article published as Journal of Algebra and Its Applications, 15, 1, 2016, 1650007. DOI:10.1142/S0219498816500079 © World Scientific Publishing Company https://www.worldscientific.com/doi/abs/10.1142/S0219498816500079, In this paper, we find the ω-value of the generators of any numerical semigroup with embedding dimension three. This allows us to determine all possible orderings of the ω-values of the generators. In addition, we relate the ω-value of the numerical semigroup to its catenary degree., The first and third authors received National Science Foundation support under DMS-1262897. The second author is supported by the projects MTM2010-15595, FQM-343, FQM-5849, NSF-1061366 and FEDER funds.

Given a numerical semigroup $S$, we let $\mathrm P_S(x)=(1-x)\sum_{s\in S}x^s$ be its semigroup polynomial. We study cyclotomic numerical semigroups; these are numerical semigroups $S$ such that $\mathrm P_S(x)$ has all its roots in the unit disc. We conjecture that $S$ is a cyclotomic numerical semigroup if and only if $S$ is a complete intersection numerical semigroup and present some evidence for it. Aside from the notion of cyclotomic numerical semigroup we introduce the notion of cyclotomic exponents and polynomially related numerical semigroups. We derive some properties and give some applications of these new concepts., 17 pages, accepted for publication in SIAM J. Discrete Math

We give an overview of the existing algorithms to compute nonunique factorization invariants in finitely generated monoids.

In this chapter we introduce the basic notions related to numerical semigroups. Numerical semigroups have not been always been referred to as such. In the past some authors called them semimodules, or demimodules and recently many authors (mainly those concerned with factorization properties) are starting to refer to them as numerical monoids.

The pseudo-Frobenius numbers of a numerical semigroup are those gaps of the numerical semigroup that are maximal for the partial order induced by the semigroup. We present a procedure to detect if a given set of integers is the set of pseudo-Frobenius numbers of a numerical semigroup and, if so, to compute the set of all numerical semigroups having this set as set of pseudo-Frobenius numbers., 22 pages

Questions concerning the lengths of factorizations into irreducible elements in numerical monoids have gained much attention in the recent literature. In this note, we show that a numerical monoid has an element with two different irreducible factorizations of the same length if and only if its embedding dimension is greater than two. We find formulas in embedding dimension three for the smallest element with two different irreducible factorizations of the same length and the largest element whose different irreducible factorizations all have distinct lengths. We show that these formulas do not naturally extend to higher embedding dimensions.

Let S be a numerical semigroup. Then there exist infinitely many symmetric numerical semigroups such that . We give an explicit description of them.

The set of solutions of the inequality axmodb⩽cx is a numerical semigroup. We present in this paper a tool for finding the set of minimal generators of this set, and thus the set of solutions to such an inequality. This tool will also enable us to give characterizations of those numerical semigroups that are the set of integer solutions of inequalities of this form. Finally, we give a deeper study of the embedding dimension three case.

The set of integer solutions to the inequality ax mod b≤c x is a numerical semigroup. We study numerical semigroups that are intersections of these numerical semigroups. Recently it has been shown that this class of numerical semigroups coincides with the class of numerical semigroups having a Toms decomposition.

Minimum distance diagrams, also known as L-shapes, have been used to study some properties related to weighted Cayley digraphs of degree two and embedding dimension three numerical semigroups. In this particular case, it has been shown that these discrete structures have at most two related L-shapes. These diagrams are proved to be a good tool for studying factorizations and the catenary degree for semigroups and diameter and distance between vertices for digraphs.; This maximum number of L-shapes has not been proved to be kept when increasing the degree of digraphs or the embedding dimension of semigroups. In this work we give a family of embedding dimension four numerical semigroups S-n, for odd n >= 5, such that the number of related L-shapes is n+3/2. This family has her analogue to weighted Cayley digraphs of degree three.; Therefore, the number of L-shapes related to numerical semigroups can be as large as wanted when the embedding dimension is at least four. The same is true for weighted Cayley digraphs of degree at least three. This fact has several implications on the combinatorics of factorizations for numerical semigroups and minimum paths between vertices for weighted digraphs. (C) 2015 Elsevier B.V. All rights reserved.

We introduce the notion of pattern for numerical semigroups, which allows us to generalize the definition of Arf numerical semigroups. In this way infinitely many other classes of numerical semigroups are defined giving a classification of the whole set of numerical semigroups. In particular, all semigroups can be arranged in an infinite non-stabilizing ascending chain whose first step consists just of the trivial semigroup and whose second step is the well-known class of Arf semigroups. We describe a procedure to compute the closure of a numerical semigroup with respect to a pattern. By using the concept of system of generators associated to a pattern, we construct recursively a directed acyclic graph with all the semigroups admitting the pattern.

This paper gives a solution to the Diophantine Frobenius problem for pseudo-symmetric numerical semigroups with embedding dimension three. We also describe a presentation for this kind of semigroup and a formula to determine whether or not a numerical semigroup with embedding dimension three is pseudo-symmetric.

Letg be a positive integer. We prove that there are positive integers n1, n2, n3 and n4 such that the semigroup S=(n1, n2, n3, n4) is an irreducible (symmetric or pseudosymmetric) numerical semigroup with g(S)=g.

For a numerical semigroup, we introduce the concept of a fundamental gap with respect to the multiplicity of the semigroup. The semigroup is fully determined by its multiplicity and these gaps. We study the case when a set of non–negative integers is the set of fundamental gaps with respect to the multiplicity of a numerical semigroup. Numerical semigroups with maximum and minimum number of this kind of gaps are described.

We introduce and study the concept of Arf system of generators for an Arf numerical semigroup. This study allows us to arrange the set of all Arf numerical semigroups in a binary tree and enables us to compute the Arf closure of a given numerical semigroup.

We study the set of numerical semigroups containing a given numerical semigroup. As an application we prove characterizations of irreducible numerical semigroups that unify some of the existing characterizations for symmetric and pseudo-symmetric numerical semigroups. Finally we describe an algorithm for computing a minimal decomposition of a numerical semigroup in terms of irreducible numerical semigroups.

We present procedures to calculate the set of Arf numerical semigroups with given genus, given conductor and given genus and conductor. We characterize the Kunz coordinates of an Arf numerical semigroup. We also describe Arf numerical semigroups with fixed Frobenius number and multiplicity up to 7.

Let n1,n2,n3 be positive integers with gcd(n1,n2,n3)=1. For S=〈n1,n2,n3〉 nonsymmetric, we give an alternative description, using elementary techniques, of a minimal presentation of its homogenization S=〈(1,0),(1,n1),(1,n2),(1,n3)〉. As a consequence, we show that this minimal presentation is unique. We recover Bresinsky’s characterization of the Cohen–Macaulay property of S and present a procedure to compute all possible catenary degrees of the elements of S.

We give an algorithm to compute the set of primitive elements for an embedding dimension three numerical semigroups. We show how we use this procedure in the study of the construction of L-shapes and the tame degree of the semigroup.

Denumerants of numerical semigroups are known to be difficult to obtain, even with small embedding dimension of the semigroups. In this work we give some results on denumerants of 3-semigroups S = 〈 a , b , c 〉 . Closed expressions are obtained under certain conditions.

Let S be a numerical semigroup. Then there exists a symmetric numerical semigroup S such that S = {n ∈ N 2n ∈ S}.

We compute the number of elements of a minimal system of generators for the congruence of a numerical semigroup with embedding dimension minus multiplicity equal to zero, one, and two. For symmetric numerical semigroups we study the cases of embedding dimension minus multiplicity equal to one, two, and three.

The weight hierarchy of one-point algebraic geometry codes can be estimated by means of the generalized order bounds, which are described in terms of a certain Weierstrass semigroup. The asymptotical behaviour of such bounds for r > 1 differs from that of the classical Feng-Rao distance (r=1) by the so-called Feng-Rao numbers. This paper is addressed to compute the Feng-Rao numbers for numerical semigroups of embedding dimension two (with two generators), obtaining a closed simple formula for the general case by using numerical semigroup techniques. These involve the computation of the Ap��ry set with respect to an integer of the semigroups under consideration. The formula obtained is applied to lower-bounding the generalized Hamming weights, improving the bound given by Kirfel and Pellikaan in terms of the classical Feng-Rao distance. We also compare our bound with a modification of the Griesmer bound, improving this one in many cases., 20 pages, 14 figures

Delorme suggested that the set of all complete intersection numerical semigroups can be computed recursively. We have implemented this algorithm, and particularized it to several subfamilies of this class of numerical semigroups: free and telescopic numerical semigroups, and numerical semigroups associated to an irreducible plane curve singularity. The recursive nature of this procedure allows us to give bounds for the embedding dimension and for the minimal generators of a semigroup in any of these families., Comment: 13 pages Results improved, References added

Patterns on numerical semigroups are multivariate linear polynomials, and they are said to admit a numerical semigroup if evaluating the pattern at any nonincreasing sequence of elements of the semigroup gives integers belonging to the semigroup. In a first approach, only homogeneous patterns were analyzed. In this contribution we study conditions for a nonhomogeneous pattern to admit a nontrivial numerical semigroup, and particularize this study to the case the independent term of the pattern is a multiple of the multiplicity of the semigroup. Moreover, for the so-called strongly admissible patterns, the set of numerical semigroups admitting these patterns with fixed multiplicity m forms an m-variety, which allows us to represent this set in a tree and to describe minimal sets of generators of the semigroups in the variety with respect to the pattern. Furthermore, we characterize strongly admissible patterns having a finite associated tree.

We give some general results concerning the computation of the generalized Feng-Rao numbers of numerical semigroups. In the case of a numerical semigroup generated by an interval, a formula for the $r^{th}$ Feng-Rao number is obtained., 23 pages, 6 figures

The concept of Frobenius variety was introduced in [67] in order to unify most of the results appearing in [88, 89, 24, 11].

If K[V] is the ring of coordinates of a variety V, then V is said to be a complete intersection if its defining ideal is generated by the least possible number of polynomials. In the special case K[V] is taken to be a semigroup ring K[S], the generators of its defining ideal can be chosen to be binomials whose exponents correspond to a presentation of the monoid S (see [41]). In this way the concept of complete intersection translates to finitely generated monoids as those having the least possible number of relations in their minimal presentations.

A generalization of the linear Diophantine Frobenius problem can be stated as follows. Let n 1 ,…,n p and d be positive integers with {n 1 ,…,n p }=1. Find a formula for the largest multiple of d not belonging to 〈n 1 ,…,n p 〉.

The aim of this chapter is to study which properties a monoid must fulfill in order to be isomorphic to a numerical semigroup. Levin shows in [46] that if S is finitely generated, Archimedean and without idempotents, then S is multiple joined. By using this as starting point, we will show that a monoid is isomorphic to a numerical semigroup if and only if it is finitely generated, quasi-Archimedean, torsion free and with only one idempotent. We will also relate this characterization with other interesting properties in semigroup theory such as weak cancellativity, being free of units, and being hereditarily finitely generated.

Herzog in [14] proves that for embedding dimension three numerical semigroups, the concepts of symmetric and complete intersection coincide. Hence with the help of Proposition 1.17 and what we know about embedding dimension two numerical semigroups, a formula for the Frobenius number and the genus of a symmetric numerical semigroup with embedding dimension three can easily be found. As for the pseudo-symmetric case, an expression for the Frobenius number of a numerical semigroup of embedding dimension three can be given in terms of the generators (and consequently also a formula for the genus in view of Corollary 3.5). This formula is presented by the authors in [82].

Redei in [53] shows that every congruence on ℕn is finitely generated. This result has since been known as Redei’s theorem, and it is equivalent to the fact that every finitely generated (commutative) monoid is finitely presented. Redei’s proof is long and elaborated. Many other authors have given alternative and much simpler proofs than his (see for instance [31, 39, 41, 56]). Since numerical semigroups are cancellative monoids, a different approach can be chosen to prove Redei’s theorem. And this is precisely the path we choose in this chapter.

We introduce the concept of fundamental gap of a numerical semigroup. We give lower and upper bounds for the number of fundamental gaps of a numerical semigroup in terms of its Frobenius number. Finally we derive several applications from the properties of fundamental gaps. © 2003 Elsevier B.V. All rights reserved.

Let n1, . . . , np be positive integers with greatest common divisor (gcd for short) one. Then it is not hard to show that there are finitely many nonnegative integers that cannot be expressed as a nonnegative integer linear combination of n1, . . . , np. The largest nonnegative integer fulfilling this condition is usually known as the Frobenius number of n1, . . . , np and it will be denoted throughout this paper by F(n1, . . . , np). The problem of determining F(n1, . . . , np) appears in the literature as the Frobenius coin-exchange problem (for a complete survey on this problem see [5], [6]). For p = 2, Sylvester proved in [7] that F(n1, n2) = n1n2 −n1 −n2. No general formula has been found so far for the case p ≥ 3. Moreover, as Curtis shows in [1], there is no closed formula of a certain type for p = 3. If we focus our attention on this case, then we can think of F as a correspondence that maps three relatively prime integers n1, n2, n3 to a nonnegative integer F(n1, n2, n3). In this paper we prove that this map is surjective, that is, for every positive integer g there exist positive integers n1, n2 and n3 such that F(n1, n2, n3) = g (Theorem 1.11). One easily realizes that every odd positive integer g can be expressed as F(2, g + 2) (see 1.1 (i)). However no even positive integers can be expressed as the Frobenius number associated to a pair of relatively prime numbers. This follows easily from Sylvester’s formula given above, since this formula can be rewritten as F(n1, n2) = (n1 − 1)(n2 − 1) − 1 and n1, n2 are coprime (in fact, it is well known that every numerical semigroup generated by two elements is symmetric, see for instance [2], [4], and thus its Frobenius number must be odd). Thus in order to express any nonnegative integer as the Frobenius number of a sequence of positive integers, the sequences used should be at least of length

Producción Científica, We establish a one-to-one correspondence between numerical semigroups of genus g and almost symmetric numerical semigroups with Frobenius number F and type F−2g, provided that F is greater than or equal to 4g−1., Ministerio de Economía, Industria y Competitividad - Fondo Europeo de Desarrollo Regional (projects MTM2017-84890-P / MTM2015-65764-C3-1-P)

Let us consider a $3$-numerical semigroup $S=\langle{a,b,N}\rangle$. Given $m\in S$, the triple $(x,y,z)\in\mathbb{N}^3$ is a factorization of $m$ in $S$ if $xa+yb+zN=m$. This work is focused on finding the full set of factorizations of any $m\in S$ and as an application we compute the catenary degree of $S$. To this end, we relate a 2D tessellation to $S$ and we use it as a main tool.

Given a numerical semigroup S ( A ) , generated by A = { a , b , N } ⊂ N with 0 a b N and gcd ( a , b , N ) = 1 , we give a parameterization of the set F ( m ; A ) = { ( x , y , z ) ∈ N 3 | x a + y b + z N = m } for any m ∈ S ( A ) . We also give the catenary degree of S ( A ) , c ( A ) . Boths results need the computation of an L-shaped tile, related to the set A, that has time-complexity O ( log N ) in the worst case.