Alberto Vigneron-Tenorio

Indispensable binomials in semigroup ideals

In this paper, we deal with the problem of the uniqueness of a minimal system of binomial generators of a semigroup ideal. Concretely, we give different necessary and/or sufficient conditions for the uniqueness of such a minimal system of generators. These conditions come from the study and combinatorial description of the so-called indispensable binomials in the semigroup ideal.

Semigroup ideals and linear diophantine equations

Abstract We give a purely algebraic algorithm to calculate the ideal of a semigroup with torsion. As application and using Grobner bases, we provide an algorithm to determine whether a linear system of equations with integer coefficients having some of the equations in congruences admits non-negative integer solutions.

Simplicial complexes and minimal free resolution of monomial algebras

Abstract This paper is concerned with the combinatorial description of the graded minimal free resolution of certain monomial algebras which includes toric rings. We explicitly describe how the graded minimal free resolution of those algebras is related to the combinatorics of some simplicial complexes. Our description may be interpreted as an algorithmic procedure to partially compute this resolution.

FIRST SYZYGIES OF TORIC VARIETIES AND DIOPHANTINE EQUATIONS IN CONGRUENCE

We compute the first syzygies of a subclass of lattice ideals by means of some abstract simplicial complexes. This subclass includes the ideals defining toric varieties. A finite check set containing the minimal first syzygy degrees is determined, and a singly-exponential bound for these degrees is explicited. Integer Programming techniques are used, precisely the Hilbert bases for diophantine equations in congruences. *Supported by Plan Propio de investigación de la Universidad de Sevilla (75403699-98-191). Fax: + 34 95 4556938; E-mail: pilar@algebra.us.es † Partially supported by Plan Propio de investigación de la Universidad de Sevilla (75403699-98-191) and Plan Propio de la Universidad de Cádiz. Fax: + 34 956 345104; E-mail: alberto.vigneron@uca.es

Affine convex body semigroups

In this paper we present a new class of semigroups called convex body semigroups which are generated by convex bodies of ℝk. They generalize to arbitrary dimension the concept of proportionally modular numerical semigroups of Rosales et al. (J. Number Theory 103, 281–294, 2003). Several properties of these semigroups are proven. Affine convex body semigroups obtained from circles and polygons of ℝ2 are characterized. The algorithms for computing minimal system of generators of these semigroups are given. We provide the implementation of some of them.

On Lawrence semigroups

Lawrence semigroups arise as a tool to compute Graver bases of semigroup ideals. It is known that the minimal free resolution of semigroup ideals is characterized by the reduced homologies of certain simplicial complexes. In this paper we study the minimal degrees of a Lawrence semigroup ideal and its first syzygy given a combinatorial characterization of the nonvanishing cycles in their associated reduced homologies. We specialize the results that appeared in [Briales, E., Campillo, A., Marijuan, C., Pison, P., 1998. Minimal systems of generators for ideals of semigroups. J. Pure Appl. Algebra, 127, 7-30] and [Pison-Casares, P., Vigneron-Tenorio, A., 2001. First syzygies of toric varieties and diophantine equations in congruence. Comm. Alg. 29 (4), 1445-1466] to the Lawrence semigroups.

Minimal Resolutions of Lattice Ideals and Integer Linear Programming

A combinatorial description of the minimal free resolution of a lattice ideal allows us to the connection of Integer Linear Programming and Al1gebra. The non null reduced homology spaces of some simplicial complexes are the key. The extremal rays of the associated cone reduce the number of variables.

An extension of Wilf’s conjecture to affine semigroups

Let

On decomposable semigroups and applications

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