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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)

- Anargyros Katsabekis, Apostolos Thoma
- J. Comb. Theory, Ser. A
- 2007

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.

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)

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)

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