Stanley–Reisner rings and the radicals of lattice ideals

In this article we associate to every lattice ideal IL,ρ⊂K[x1,…,xm]IL,ρ⊂K[x1,…,xm] a cone σσ and a simplicial complex ΔσΔσ with vertices the minimal generators of the Stanley–Reisner ideal of σσ. We assign a simplicial subcomplex Δσ(F)Δσ(F) of ΔσΔσ to every polynomial F  . If F1,…,FsF1,…,Fs generate IL,ρIL,ρ or they generate rad(IL,ρ)rad(IL,ρ) up to radical, then ⋃i=1sΔσ(Fi) is a spanning subcomplex of ΔσΔσ. This result provides a lower bound for the minimal number of generators of IL,ρIL,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.