On the binomial arithmetical rank of 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.