Stanley–Reisner rings and the radicals of lattice ideals

AbstractIn this article we associate to every lattice ideal IL,ρ⊂K[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 ⋃i=1sΔσ(Fi) 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 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