Stanley-Reisner rings and the radicals of lattice ideals

In this article we associate to every lattice ideal I_L,ρ⊂Kx₁,…,x_m] a cone σ and a simplicial complex Δ_σ with vertices the minimal generators of the Stanley-Reisner ideal of σ. We assign a simplicial subcomplex Δ_σ(F) of Δ_σ to every polynomial F. If F₁,…,F_s generate I_L,ρ or they generate rad(I_L,ρ) up to radical, then is a spanning subcomplex of Δ_σ. This result provides a lower bound for the minimal number of generators of I_L,ρ which 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 given bounds are sharp.