559 research outputs found
Complete intersection vanishing ideals on degenerate tori over finite fields
We study the complete intersection property and the algebraic invariants
(index of regularity, degree) of vanishing ideals on degenerate tori over
finite fields. We establish a correspondence between vanishing ideals and toric
ideals associated to numerical semigroups. This correspondence is shown to
preserve the complete intersection property, and allows us to use some
available algorithms to determine whether a given vanishing ideal is a complete
intersection. We give formulae for the degree, and for the index of regularity
of a complete intersection in terms of the Frobenius number and the generators
of a numerical semigroup.Comment: Arabian Journal of Mathematics, to appea
Complete intersections in binomial and lattice ideals
For the family of graded lattice ideals of dimension 1, we establish a
complete intersection criterion in algebraic and geometric terms. In positive
characteristic, it is shown that all ideals of this family are binomial set
theoretic complete intersections. In characteristic zero, we show that an
arbitrary lattice ideal which is a binomial set theoretic complete intersection
is a complete intersection.Comment: Internat. J. Algebra Comput., to appea
Computing the degree of a lattice ideal of dimension one
We show that the degree of a graded lattice ideal of dimension 1 is the order
of the torsion subgroup of the quotient group of the lattice. This gives an
efficient method to compute the degree of this type of lattice ideals.Comment: J. Symbolic Comput., to appea
Binomial Ideals and Congruences on Nn
Producción CientíficaA congruence on Nn is an equivalence relation on Nn that is compatible with the additive structure. If k is a field, and I is a binomial ideal in k[X1,…,Xn] (that is, an ideal generated by polynomials with at most two terms), then I induces a congruence on Nn by declaring u and v to be equivalent if there is a linear combination with nonzero coefficients of Xu and Xv that belongs to I. While every congruence on Nn arises this way, this is not a one-to-one correspondence, as many binomial ideals may induce the same congruence. Nevertheless, the link between a binomial ideal and its corresponding congruence is strong, and one may think of congruences as the underlying combinatorial structures of binomial ideals. In the current literature, the theories of binomial ideals and congruences on Nn are developed separately. The aim of this survey paper is to provide a detailed parallel exposition, that provides algebraic intuition for the combinatorial analysis of congruences. For the elaboration of this survey paper, we followed mainly (Kahle and Miller Algebra Number Theory 8(6):1297–1364, 2014) with an eye on Eisenbud and Sturmfels (Duke Math J 84(1):1–45, 1996) and Ojeda and Piedra Sánchez (J Symbolic Comput 30(4):383–400, 2000).National Science Foundation (grant DMS-1500832)Ministerio de Economía, Industria y Competitividad (project MTM2015-65764-C3-1)Junta de Extremadura (grupo de investigación FQM-024
Minimal Castelnuovo-Mumford regularity for a given Hilbert polynomial
Let be an algebraically closed field of null characteristic and a
Hilbert polynomial. We look for the minimal Castelnuovo-Mumford regularity
of closed subschemes of projective spaces over with Hilbert
polynomial . Experimental evidences led us to consider the idea that
could be achieved by schemes having a suitable minimal Hilbert
function. We give a constructive proof of this fact. Moreover, we are able to
compute the minimal Castelnuovo-Mumford regularity of
schemes with Hilbert polynomial and given regularity of the
Hilbert function, and also the minimal Castelnuovo-Mumford regularity of
schemes with Hilbert function . These results find applications in the study
of Hilbert schemes. They are obtained by means of minimal Hilbert functions and
of two new constructive methods which are based on the notion of
growth-height-lexicographic Borel set and called ideal graft and extended
lifting.Comment: 21 pages. Comments are welcome. More concise version with a slight
change in the title. A further revised version has been accepted for
publication in Experimental Mathematic
- …