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In this article we associate to every lattice ideal I L,ρ ⊂ K[x 1 ,. .. , x m ] a cone σ and a graph G σ with vertices the minimal generators of the Stanley-Reisner ideal of σ. To every polynomial F we assign a subgraph G σ (F) of the graph G σ. Every expression of the radical of I L,ρ , as a radical of an ideal generated by some polynomials F 1 ,. .. , F s… (More)

Let A = {a 1 ,. .. , a m } ⊂ Z n be a vector configuration and I A ⊂ K[x 1 ,. .. , x m ] 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. In the second part we associate to A a simplicial complex ∆ ind(A). We show that the vertices… (More)

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.

Let I L,ρ be a lattice ideal. We provide a necessary and sufficient criterion under which a set of binomials in I L,ρ generate the radical of I L,ρ up to radical. We apply our results to the problem of determining the minimal number of generators of I L,ρ or of the lattice ideal rad(I L,ρ) up to radical.

In this article we study specializations of multigradings and apply them to the problem of the computation of the arithmetical rank of a lattice ideal IL G ⊂ K[x1,. .. , xn]. The arithmetical rank of IL G equals the F-homogeneous arithmetical rank of IL G , for an appropriate specialization F of G. To the lattice ideal IL G and every specialization F of G… (More)

Let I G ⊂ K[x 1 ,. .. , xm] be the toric ideal associated to a finite graph G. In this paper we study the binomial arithmetical rank and the G-homogeneous arithmetical rank of I G in 2 cases: (1) G is bipartite, (2) I G is generated by quadratic binomials. In both cases we prove that the binomial arithmetical rank and the G-arithmetical rank coincide with… (More)

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… (More)

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