Stanley-Reisner rings and the radicals of lattice ideals

Abstract

In this article we associate to every lattice ideal $I_{L,\rho}\subset K[x_1,..., x_m]$ a cone $\sigma $ and a graph $G_{\sigma}$ with vertices the minimal generators of the Stanley-Reisner ideal of $\sigma $. To every polynomial $F$ we assign a subgraph $G_{\sigma}(F)$ of the graph $G_{\sigma}$. Every expression of the radical of $I_{L,\rho}$, as a radical of an ideal generated by some polynomials $F_1,..., F_s$ gives a spanning subgraph of $G_{\sigma}$, the $\cup_{i=1}^s G_{\sigma}(F_i)$. This result provides a lower bound for the minimal number of generators of $I_{L,\rho}$ and therefore improves the generalized Krull's principal ideal theorem for lattice ideals. But mainly it provides lower bounds for the binomial arithmetical rank and the $A$-homogeneous arithmetical rank of a lattice ideal. Finally we show, by a family of examples, that the bounds given are sharp.