Specializations of multigradings and the arithmetical rank of lattice ideals

TL;DR

This paper explores how specializations of multigradings can be used to analyze the arithmetical rank of lattice ideals, linking algebraic properties to combinatorial invariants of associated simplicial complexes.

Contribution

It introduces a method to compute the arithmetical rank of lattice ideals via specializations and simplicial complexes, providing new lower bounds based on combinatorial invariants.

Findings

Arithmetical rank equals the $ ext{F}$-homogeneous arithmetical rank for suitable specializations.

Associates a simplicial complex to each lattice ideal and specialization.

Combinatorial invariants of the complex give lower bounds for the arithmetical rank.

Abstract

In this article we study specializations of multigradings and apply them to the problem of the computation of the arithmetical rank of a lattice ideal $I_{L_{\mathcal{G}}} \subset K[x_{1},...,x_{n}]$. The arithmetical rank of $I_{L_{\mathcal{G}}}$ equals the $\mathcal{F}$-homogeneous arithmetical rank of $I_{L_{\mathcal{G}}}$, for an appropriate specialization $\mathcal{F}$ of $\mathcal{G}$. To the lattice ideal $I_{L_{\mathcal{G}}}$ and every specialization $\mathcal{F}$ of $\mathcal{G}$ we associate a simplicial complex. We prove that combinatorial invariants of the simplicial complex provide lower bounds for the $\mathcal{F}$-homogeneous arithmetical rank of $I_{L_{\mathcal{G}}}$.