Minimal Systems of Binomial Generators and the Indispensable Complex of a Toric Ideal

Let A = {a1, . . . , am} ⊂ Z be a vector configuration and IA ⊂ K[x1, . . . , xm] its corresponding toric ideal. 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. 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 IA, while one dimensional facets of ∆ind(A) with minimal binomial A-degree correspond to the indispensable binomials of IA. This talk is based on joint work with A. Katsabekis and A. Thoma.