Degree and algebraic properties of lattice and matrix ideals

TL;DR

This paper investigates the algebraic and geometric properties of lattice and matrix ideals, providing formulas for their degrees, analyzing their primary decompositions, and exploring applications to graph theory and binomial ideals.

Contribution

It introduces new formulas for degrees of lattice ideals, studies their primary decompositions over arbitrary fields, and extends properties of binomial matrix ideals, including applications to graph invariants.

Findings

Degree formulas involving torsion and lattice polytopes

Closure of GPCB matrices under transposition

Degree of Laplacian ideal equals sandpile group order

Abstract

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.