Search

Let S⊆N be a numerical semigroup with multiplicity m, conductor c and minimal generating set P. Let L=S∩[0,c-1] and W(S)=|P||L|-c. In 1978, Herbert Wilf asked whether W(S)≥0 always holds, a question known as Wilf's conjecture and open since then. A related number W0(S), satisfying W0(S)≤W(S), has recently been introduced. We say that S is a near-miss in Wilf's conjecture if W0(S)<0. Near-misses are very rare. Here we construct infinite families of them, with c=4m and W0(S) arbitrarily small, and we show that the members of these families still satisfy Wilf's conjecture. [ABSTRACT FROM AUTHOR]

This paper is focused on numerical semigroups and presents a simple construction, that we call "dilatation", which, from a starting semigroup S, permits to get an infinite family of semigroups which share several properties with S. The invariants of each semigroup T of this family are given in terms of the corresponding invariants of S and both the Apéry set and the minimal generators of T are described. We also study three properties that are close to the Gorenstein property of the associated semigroup ring: almost Gorenstein, 2-AGL, and nearly Gorenstein properties. More precisely, we prove that S satisfies one of these properties if and only if each dilatation of S satisfies the corresponding one. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

The common behaviour of many families of numerical semigroups led to defining, firstly, the Frobenius varieties and, secondly, the (Frobenius) pseudo-varieties. However, some interesting families are still not covered by these definitions. To overcome this situation, here we introduce the concept of Frobenius restricted variety (or R-variety). We generalize most of the results for varieties and pseudo-varieties to R-varieties. In particular, we study the tree structure that arises within them. [ABSTRACT FROM AUTHOR]

We introduce the concept of homogeneous numerical semigroups and show that all homogeneous numerical semigroups with Cohen-Macaulay tangent cones are of homogeneous type. In embedding dimension three, we classify all numerical semigroups of homogeneous type into numerical semigroups with complete intersection tangent cones and the homogeneous ones which are not symmetric with Cohen-Macaulay tangent cones. We also study the behavior of the homogeneous property by gluing and shiftings to construct large families of homogeneous numerical semigroups with Cohen-Macaulay tangent cones. In particular we show that these properties fulfill asymptotically in the shifting classes. Several explicit examples are provided along the paper to illustrate the property. [ABSTRACT FROM AUTHOR]

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 Apéry 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. [ABSTRACT FROM AUTHOR]

This article introduces patterns of ideals of numerical semigroups, thereby unifying previous definitions of patterns of numerical semigroups. Several results of general interest are proved. More precisely, this article presents results on the structure of the image of patterns of ideals, and also on the structure of the sets of patterns admitted by a given ideal. [ABSTRACT FROM AUTHOR]

Given positive integers $$n_1,\ldots ,n_p$$ , we say that a submonoid M of $$({\mathbb N},+)$$ is a $$(n_1,\ldots ,n_p)$$ -bracelet if $$a+b+\left\{ n_1,\ldots ,n_p\right\} \subseteq M$$ for every $$a,b\in M\backslash \left\{ 0\right\} $$ . In this note, we explicitly describe the smallest $$\left( n_1,\ldots ,n_p\right) $$ -bracelet that contains a finite subset X of $${\mathbb N}$$ . We also present a recursive method that enables us to construct the whole set $$\mathcal B(n_1,\ldots ,n_p)=\left\{ M|M \quad \text {is a} \quad (n_1,\ldots ,n_p)\text {-bracelet}\right\} $$ . Finally, we study $$(n_1,\ldots ,n_p)$$ -bracelets that cannot be expressed as the intersection of $$(n_1,\ldots , n_p)$$ -bracelets properly containing it. [ABSTRACT FROM AUTHOR]

In this paper we present an algorithm for computing the set of saturated numerical semigroups of a given genus. We see how the set of saturated numerical semigroups can be arranged in a tree rooted in $\mathbb{N}$ and we describe the sons of any vertex of this tree. [ABSTRACT FROM AUTHOR]

We study arithmetic properties of tangent cones associated to large families of monomial curves obtained by gluing. In particular, we characterize their Cohen-Macaulay and Gorenstein properties and prove that they have non-decreasing Hilbert functions. The results come from a careful analysis of some special Apéry sets of the numerical semigroups obtained by gluing under a condition that we call specific gluing. As a consequence, we complete and extend the results proved by Arslan et al. (in Proc. Am. Math. Soc. 137:2225-2232, ) about nice gluings by using different techniques. Our results also allow to prove that for a given numerical semigroup with a non-decreasing Hilbert function and an integer q>1, all extensions of it by q, except a finite number, have non-decreasing Hibert functions. [ABSTRACT FROM AUTHOR]

We prove that the numerical semigroups associated to the combinatorial configurations satisfy a family of linear, non-homogeneous, symmetric patterns. We use these patterns to prove an upper bound of the conductor and we also give an upper bound of the multiplicity. Also, we compare bounds of the conductor of numerical semigroups associated to balanced configurations, and to configurations with coprime parameters. The proof of the latter involves a bound of the conductor of prime power generated numerical semigroups. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]