Stanley-Reisner rings and the radicals of lattice ideals

In this article we associate to every lattice ideal I(L,p) subset of K[x(1),..., x(m)] a cone a and a simplicial complex Delta(sigma) with vertices the minimal generators of the Stanley-Reisner ideal of a. We assign a simplicial subcomplex A(sigma)(F) of A(sigma) to every polynomial F. If F(1)..., F(s) generate I(L,p) or they generate rad(I(L,p)) up to radical, then boolean OR(s)(i=l) Delta(sigma)(F(i)) is a spanning subcomplex of A. This result provides a lower bound for the minimal number of generators of I(L,p) 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. (c) 2005 Elsevier B.V. All rights reserved.