Stanley–Reisner rings and the radicals of lattice ideals

In this article we associate to every lattice ideal IL,ρsubset ofK[x1,…,xm] a cone σ and a simplicial complex Δσ with vertices the minimal generators of the Stanley–Reisner ideal of σ. We assign a simplicial subcomplex Δσ(F) of Δσ to every polynomial F. If F1,…,Fs generate IL,ρ or they generate rad(IL,ρ) up to radical, then image is a spanning subcomplex of Δσ. This result provides a lower bound for the minimal number of generators of IL,ρ which improves the generalized Krullʹs 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.