Indispensable binomials in semigroup ideals

In this paper, we deal with the problem of uniqueness of minimal system of binomial generators of a semigroup ideal. Concretely, we give different necessary and/or sufficient conditions for uniqueness of such minimal system of generators. These conditions come from the study and combinatorial description of the so-called indispensable binomials in the semigroup ideal.Comment: 11 pages. This paper was initially presented at the II Iberian Mathematical Meeting (http://imm2.unex.es). To appear in the Proc. Amer. Math. So

The short resolution of a semigroup algebra

This work generalizes the short resolution given in Proc. Amer. Math. Soc. \textbf{131}, 4, (2003), 1081--1091, to any affine semigroup. Moreover, a characterization of Ap\'{e}ry sets is given. This characterization lets compute Ap\'{e}ry sets of affine semigroups and the Frobenius number of a numerical semigroup in a simple way. We also exhibit a new characterization of the Cohen-Macaulay property for simplicial affine semigroups.Comment: 12 pages. In this new version, some proofs have been detailed, the references on the computatation of the Frobenius number of a numerical semigroup have been updated and some typpos have been correcte

Uniquely presented finitely generated commutative monoids

A finitely generated commutative monoid is uniquely presented if it has only a minimal presentation. We give necessary and sufficient conditions for finitely generated, combinatorially finite, cancellative, commutative monoids to be uniquely presented. We use the concept of gluing to construct commutative monoids with this property. Finally for some relevant families of numerical semigroups we describe the elements that are uniquely presented.Comment: 13 pages, typos corrected, references update

Uniqueness of limit cycles for quadratic vector fields

Producción CientíficaThis article deals with the study of the number of limit cycles surrounding a critical point of a quadratic planar vector field, which, in normal form, can be written as x ′ = a1x − y − a3x 2 + (2a2 + a5)xy+a6y 2 , y ′ = x+a1y+a2x 2+(2a3+a4)xy−a2y 2 . In particular, we study the semi-varieties defined in terms of the parameters a1, a2, . . . , a6 where some classical criteria for the associated Abel equation apply. The proofs will combine classical ideas with tools from computational algebraic geometry.Agencia Estatal de Investigación - Fondo Europeo de Desarrollo Regional (grant MTM 2011-22751)Junta de Extremadura (grant GR15055)Ministerio de Economía, Industria y Competitividad - Fondo Europeo de Desarrollo Regional (grant MTM2015-65764-C3-1-P

An indispensable classification of monomial curves in \mathbb{A}^4(\mathbbmss{k})

In this paper a new classification of monomial curves in \mathbb{A}^4(\mathbbmss{k}) is given. Our classification relies on the detection of those binomials and monomials that have to appear in every system of binomial generators of the defining ideal of the monomial curve; these special binomials and monomials are called indispensable in the literature. This way to proceed has the advantage of producing a natural necessary and sufficient condition for the definining ideal of a monomial curve in \mathbb{A}^4(\mathbbmss{k}) to have a unique minimal system of binomial generators. Furthermore, some other interesting results on more general classes of binomial ideals with unique minimal system of binomial generators are obtained.Comment: 17 pages; fixed typos, added some clarifying remarks, minor corrections to the original version. Accepted for publication in Pacific Journal of Mathematic