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Search Results: 1 - 10 of 683141 matches for " Pedro A. García-Sánchez "
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A new approach for the computation of the tame degree
Pedro A. García-Sánchez
Mathematics , 2015,
Abstract: In this paper we present a new procedure to compute the tame degree of an atomic monoid that gives rise to a faster algorithm for full affine semigroups.
On curves with one place at infinity
Abdallah Assi,Pedro A. García-Sánchez
Mathematics , 2014,
Abstract: Let $f$ be a plane curve. We give a procedure based on Abhyankar's approximate roots to detect if it has a single place at infinity, and if so construct its associated $\delta$-sequence, and consequently its value semigroup. Also for fixed genus (equivalently Frobenius number) we construct all $\delta$-sequences generating numerical semigroups with this given genus. For a $\delta$-sequence we present a procedure to construct all curves having this associated sequence. We also study the embeddings of such curves in the plane. In particular, we prove that polynomial curves might not have a unique embedding.
Numerical semigroups and applications
Abdallah Assi,Pedro A. García-Sánchez
Mathematics , 2014,
Abstract: The aim of this manuscript is to give some basic notions related to numerical semigroups, and from these on the one hand describe a classical application to the study of singularities of plane algebraic curves, and on the other, show how numerical semigroups can be used to obtain handy examples of nonunique factorization invariants.
Constructing the set of complete intersection numerical semigroups with a given Frobenius number
Abdallah Assi,Pedro A. García-Sánchez
Mathematics , 2012,
Abstract: 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.
Factorization invariants in half-factorial affine semigroups
Pedro A. García-Sánchez,Ignacio Ojeda,Alfredo Sánchez-R. -Navarro
Mathematics , 2012, DOI: 10.1142/S0218196713500033
Abstract: Let $\mathbb{N} \mathcal{A}$ be the monoid generated by $\mathcal{A} = {\mathbf{a}_1, ..., \mathbf{a}_n} \subseteq \mathbb{Z}^d.$ We introduce the homogeneous catenary degree of $\mathbb{N} \mathcal{A}$ as the smallest $N \in \mathbb N$ with the following property: for each $\mathbf{a} \in \mathbb{N} \mathcal{A}$ and any two factorizations $\mathbf{u}, \mathbf{v}$ of $\mathbf{a}$, there exists factorizations $\mathbf{u} = \mathbf{w}_1, ..., \mathbf{w}_t = \mathbf{v} $ of $\mathbf{a}$ such that, for every $k, \mathrm{d}(\mathbf{w}_k, \mathbf{w}_{k+1}) \leq N,$ where $\mathrm{d}$ is the usual distance between factorizations, and the length of $\mathbf{w}_k, |\mathbf{w}_k|,$ is less than or equal to $\max{|\mathbf{u}|, |\mathbf{v}|}.$ We prove that the homogeneous catenary degree of $\mathbb{N} \mathcal{A}$ improves the monotone catenary degree as upper bound for the ordinary catenary degree, and we show that it can be effectively computed. We also prove that for half-factorial monoids, the tame degree and the $\omega$-primality coincide, and that all possible catenary degrees of the elements of an affine semigroup of this kind occur as the catenary degree of one of its Betti elements.
Nonhomogeneous patterns on numerical semigroups
Maria Bras-Amorós,Pedro A. García-Sánchez,Albert Vico-Oton
Mathematics , 2012,
Abstract: Patterns on numerical semigroups are multivariate linear polynomials, and they are said to be admissible if there exists a numerical semigroup such that evaluated at any nonincreasing sequence of elements of the semigroup gives integers belonging to the semigroup. In a first approach, only homogeneous patterns where analized. In this contribution we study conditions for an eventually non-homogeneous pattern to be admissible, 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 $m$ form 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.
Semigroup-theoretical characterizations of arithmetical invariants with applications to numerical monoids and Krull monoids
Víctor Blanco,Pedro A. García-Sánchez,Alfred Geroldinger
Mathematics , 2010,
Abstract: Arithmetical invariants---such as sets of lengths, catenary and tame degrees---describe the non-uniqueness of factorizations in atomic monoids. We study these arithmetical invariants by the monoid of relations and by presentations of the involved monoids. The abstract results will be applied to numerical monoids and to Krull monoids.
Frobenius vectors, Hilbert series and gluings
Abdallah Assi,Pedro A. García-Sánchez,Ignacio Ojeda
Mathematics , 2013,
Abstract: Let $S_1$ and $S_2$ be two affine semigroups and let $S$ be the gluing of $S_1$ and $S_2$. Several invariants of $S$ are then related to those of $S_1$ and $S_2$; we review some of the most important properties preserved under gluings. The aim of this paper is to prove that this is the case for the Frobenius vector and the Hilbert series. Applications to complete intersection affine semigroups are also given.
Cyclotomic numerical semigroups
Emil-Alexandru Ciolan,Pedro A. García-Sánchez,Pieter Moree
Mathematics , 2014,
Abstract: 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.
Affine semigroups having a unique Betti element
Pedro A. García-Sánchez,Ignacio Ojeda,José Carlos Rosales
Mathematics , 2012, DOI: 10.1142/S0219498812501770
Abstract: We characterize affine semigroups having one Betti element and we compute some relevant non-unique factorization invariants for these semigroups. As an example, we particularize our description to numerical semigroups.
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