Abstract. Let A = {a 1 , . . . , am} ⊂ Z n be a vector configuration and I A ⊂ K[x 1 , . . . , xm] its corresponding toric ideal. The paper consists of two parts. In the first part we completely determine the number of different minimal systems of binomial generators of I A . We also prove that generic toric ideals are generated by indispensable binomials. In the second part we associate to A a simplicial complex ∆ ind(A) . We show that the vertices of ∆ ind(A) correspond to the indispensable monomials of the toric ideal I A , while one dimensional facets of ∆ ind(A) with minimal binomial A-degree correspond to the indispensable binomials of I A .

Using a generalized notion of matching in a simplicial complex and circuits of vector configurations, we compute lower bounds for the minimum number of generators, the binomial arithmetical rank and the A-homogeneous arithmetical rank of a lattice ideal. Prime lattice ideals are toric ideals, i.e. the defining ideals of toric varieties.

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In this article we associate to every lattice ideal I L, ⊂ K[x 1 , . . . , x m ] a cone and a simplicial complex with vertices the minimal generators of the Stanley-Reisner ideal of . We assign a simplicial subcomplex (F ) of to every polynomial F. If F 1 , . . . , F s generate I L, or they generate rad(I L, ) up to radical, then s i=1 (F i ) is a spanning subcomplex of . This result provides a lower bound for the minimal number of generators of I L, 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. (A. Katsabekis), Marcel.Morales@ujf-grenoble.fr (M. Morales), athoma@cc.uoi.gr (A. Thoma).

Let I M and I N be defining ideals of toric varieties such that I M is a projection of I N , i.e. I N ⊆ I M . We give necessary and sufficient conditions for the equality. . , f s belong to I M . Also, a method for finding toric varieties which are set-theoretic complete intersection is given. Finally, we apply our method in the computation of the arithmetical rank of certain toric varieties and provide the defining equations of the above toric varieties.