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.