Binomial generation of the radical of a lattice ideal

TL;DR

This paper establishes a criterion for when binomials generate the radical of a lattice ideal and explores implications for minimal generating sets.

Contribution

It introduces a necessary and sufficient condition for binomials to generate the radical of a lattice ideal, advancing understanding of their algebraic structure.

Findings

Provided a criterion for radical generation by binomials

Applied results to minimal generator problems

Enhanced understanding of lattice ideal radicals

Abstract

Let $I_{L, \rho}$ be a lattice ideal. We provide a necessary and sufficient criterion under which a set of binomials in $I_{L, \rho}$ generate the radical of $I_{L, \rho}$ up to radical. We apply our results to the problem of determining the minimal number of generators of $I_{L, \rho}$ or of the $rad(I_{L, \rho})$ up to radical.