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In this article we associate to every lattice ideal IL,ρ ⊂ K[x1, . . . , xm] a cone σ and a graph Gσ with vertices the minimal generators of the Stanley-Reisner ideal of σ. To every polynomial F we assign a subgraph Gσ(F ) of the graph Gσ. Every expression of the radical of IL,ρ, as a radical of an ideal generated by some polynomials F1, . . . , Fs gives a… (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. © 2006 Elsevier Inc.… (More)

Let IM and IN be defining ideals of toric varieties such that IM is a projection of IN , i.e. IN ⊆ IM . We give necessary and sufficient conditions for the equality IM = rad(IN + (f1, . . . , fs)), where f1, . . . , fs belong to IM . Also a method for finding toric varieties which are set-theoretic complete intersection is given. Finally we apply our method… (More)

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

Let A = {a1, . . . ,am} ⊂ Z be a vector configuration and IA ⊂ K[x1, . . . , 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 IA. We also prove that generic toric ideals are generated by indispensable binomials. In the second part… (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 ILG ⊂ K[x1, . . . , xn]. The arithmetical rank of ILG equals the F-homogeneous arithmetical rank of ILG , for an appropriate specialization F of G. To the lattice ideal ILG and every specialization F of G we… (More)

Let IG ⊂ K[x1, . . . , xm] be the toric ideal associated to a finite graph G. In this paper we study the binomial arithmetical rank and the Ghomogeneous arithmetical rank of IG in 2 cases: (1) G is bipartite, (2) IG is generated by quadratic binomials. In both cases we prove that the binomial arithmetical rank and the G-arithmetical rank coincide with the… (More)

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