Stanley-Reisner rings and the radicals of lattice ideals

Abstract

In this article we associate to every lattice ideal $I_{L,ρ}\subset K[x_1,..., x_m]$ a cone $σ$ and a graph $G_σ$ with vertices the minimal generators of the Stanley-Reisner ideal of $σ$. To every polynomial $F$ we assign a subgraph $G_σ(F)$ of the graph $G_σ$. Every expression of the radical of $I_{L,ρ}$, as a radical of an ideal generated by some polynomials $F_1,..., F_s$ gives a spanning subgraph of $G_σ$, the $\cup_{i=1}^s G_σ(F_i)$. This result provides a lower bound for the minimal number of generators of $I_{L,ρ}$ 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.