7,795 research outputs found
Degree and algebraic properties of lattice and matrix ideals
We study the degree of non-homogeneous lattice ideals over arbitrary fields,
and give formulae to compute the degree in terms of the torsion of certain
factor groups of Z^s and in terms of relative volumes of lattice polytopes. We
also study primary decompositions of lattice ideals over an arbitrary field
using the Eisenbud-Sturmfels theory of binomial ideals over algebraically
closed fields. We then use these results to study certain families of integer
matrices (PCB, GPCB, CB, GCB matrices) and the algebra of their corresponding
matrix ideals. In particular, the family of generalized positive critical
binomial matrices (GPCB matrices) is shown to be closed under transposition,
and previous results for PCB ideals are extended to GPCB ideals. Then, more
particularly, we give some applications to the theory of 1-dimensional binomial
ideals. If G is a connected graph, we show as a further application that the
order of its sandpile group is the degree of the Laplacian ideal and the degree
of the toppling ideal. We also use our earlier results to give a structure
theorem for graded lattice ideals of dimension 1 in 3 variables and for
homogeneous lattices in Z^3 in terms of critical binomial ideals (CB ideals)
and critical binomial matrices, respectively, thus complementing a well-known
theorem of Herzog on the toric ideal of a monomial space curve.Comment: SIAM J. Discrete Math., to appea
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
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