Minimal Generating Sets of Lattice Ideals

TL;DR

This paper characterizes minimal generating sets of lattice ideals using graph constructions and solves the open problem of identifying binomial complete intersection lattice ideals for non-positive lattices.

Contribution

It provides a description of minimal binomial generating sets of lattice ideals and characterizes binomial complete intersections in non-positive cases.

Findings

Describes minimal binomial generating sets of lattice ideals.

Uses graph construction on fiber equivalence classes.

Characterizes binomial complete intersection lattice ideals.

Abstract

Let $L\subset \mathbb{Z}^n$ be a lattice and $I_L=\langle x^{\bf u}-x^{\bf v}:\ {\bf u}-{\bf v}\in L\rangle$ be the corresponding lattice ideal in $\Bbbk[x_1,\ldots, x_n]$, where $\Bbbk$ is a field. In this paper we describe minimal binomial generating sets of $I_L$ and their invariants. We use as a main tool a graph construction on equivalence classes of fibers of $I_L$. As one application of the theory developed we characterize binomial complete intersection lattice ideals, a longstanding open problem in the case of non-positive lattices.