On the binomial arithmetical rank of lattice ideals

TL;DR

This paper investigates the relationship between the binomial arithmetical rank and the minimal number of generators of lattice ideals, providing bounds and conditions for equality, and exploring properties of determinantal lattice ideals.

Contribution

It establishes lower bounds for the binomial arithmetical rank and identifies cases where it equals the minimal number of generators, also analyzing a class of determinantal lattice ideals.

Findings

Lower bounds for binomial arithmetical rank and $\\mathcal{A}$-homogeneous arithmetical rank.

Conditions under which binomial arithmetical rank equals the minimal number of generators.

Analysis of algebraic properties of determinantal lattice ideals.

Abstract

To any lattice $L \subset \mathbb{Z}^{m}$ one can associate the lattice ideal $I_{L} \subset K[x_{1},...,x_{m}]$. This paper concerns the study of the relation between the binomial arithmetical rank and the minimal number of generators of $I_{L}$. We provide lower bounds for the binomial arithmetical rank and the $\mathcal{A}$-homogeneous arithmetical rank of $I_{L}$. Furthermore, in certain cases we show that the binomial arithmetical rank equals the minimal number of generators of $I_{L}$. Finally we consider a class of determinantal lattice ideals and study some algebraic properties of them.