Stanley–Reisner rings and the radicals of lattice ideals

Abstract

In this article we associate to every lattice ideal I L , ρ ⊂ K [ x 1 , … , 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 F 1 , … , F s generate I L , ρ or they generate rad ( I L , ρ ) up to radical, then ⋃ i = 1 s Δ σ ( F i ) 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.