1,940 research outputs found
Regularity and algebraic properties of certain lattice ideals
We study the regularity and the algebraic properties of certain lattice
ideals. We establish a map I --> I\~ between the family of graded lattice
ideals in an N-graded polynomial ring over a field K and the family of graded
lattice ideals in a polynomial ring with the standard grading. This map is
shown to preserve the complete intersection property and the regularity of I
but not the degree. We relate the Hilbert series and the generators of I and
I\~. If dim(I)=1, we relate the degrees of I and I\~. It is shown that the
regularity of certain lattice ideals is additive in a certain sense. Then, we
give some applications. For finite fields, we give a formula for the regularity
of the vanishing ideal of a degenerate torus in terms of the Frobenius number
of a semigroup. We construct vanishing ideals, over finite fields, with
prescribed regularity and degree of a certain type. Let X be a subset of a
projective space over a field K. It is shown that the vanishing ideal of X is a
lattice ideal of dimension 1 if and only if X is a finite subgroup of a
projective torus. For finite fields, it is shown that X is a subgroup of a
projective torus if and only if X is parameterized by monomials. We express the
regularity of the vanishing ideal over a bipartie graph in terms of the
regularities of the vanishing ideals of the blocks of the graph.Comment: Bull. Braz. Math. Soc. (N.S.), to appea
Binomial D-modules
We study quotients of the Weyl algebra by left ideals whose generators
consist of an arbitrary Z^d-graded binomial ideal I along with Euler operators
defined by the grading and a parameter in C^d. We determine the parameters for
which these D-modules (i) are holonomic (equivalently, regular holonomic, when
I is standard-graded); (ii) decompose as direct sums indexed by the primary
components of I; and (iii) have holonomic rank greater than the generic rank.
In each of these three cases, the parameters in question are precisely those
outside of a certain explicitly described affine subspace arrangement in C^d.
In the special case of Horn hypergeometric D-modules, when I is a lattice basis
ideal, we furthermore compute the generic holonomic rank combinatorially and
write down a basis of solutions in terms of associated A-hypergeometric
functions. This study relies fundamentally on the explicit lattice point
description of the primary components of an arbitrary binomial ideal in
characteristic zero, which we derive in our companion article arxiv:0803.3846.Comment: This version is shorter than v2. The material on binomial primary
decomposition has been split off and now appears in its own paper
arxiv:0803.384
Binomial generation of the radical of a lattice ideal
Let be a lattice ideal. We provide a necessary and sufficient
criterion under which a set of binomials in generate the radical
of up to radical. We apply our results to the problem of
determining the minimal number of generators of or of the
up to radical.Comment: 14 pages, to appear in Journal of Algebr
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
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