Binomial Difference Ideals

TL;DR

This paper develops theoretical foundations and algorithms for analyzing binomial difference ideals, including their representations, properties, and closures, with applications to ideal decomposition.

Contribution

It introduces canonical representations, criteria for ideal properties, and algorithms for closure computation and decomposition of binomial difference ideals.

Findings

Provided criteria for reflexivity, primality, well-mixedness, and perfection.

Developed algorithms for property checking and closure computation.

Extended properties from Laurent to general binomial difference ideals.

Abstract

In this paper, binomial difference ideals are studied. Three canonical representations for Laurent binomial difference ideals are given in terms of the reduced Groebner basis of Z[x]-lattices, regular and coherent difference ascending chains, and partial characters over Z[x]-lattices, respectively. Criteria for a Laurent binomial difference ideal to be reflexive, prime, well-mixed, and perfect are given in terms of their support lattices. The reflexive, well-mixed, and perfect closures of a Laurent binomial difference ideal are shown to be binomial. Most of the properties of Laurent binomial difference ideals are extended to the case of difference binomial ideals. Finally, algorithms are given to check whether a given Laurent binomial difference ideal I is reflexive, prime, well-mixed, or perfect, and in the negative case, to compute the reflexive, well-mixed, and perfect closures of I. An algorithm is given to decompose a finitely generated perfect binomial difference ideal as the intersection of reflexive prime binomial difference ideals.