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 .

Let I G be the toric ideal of a graph G. We characterize in graph theoretical terms primitive, minimal, indispensable and fundamental binomials of the toric ideal I G .

To any toric ideal I A , encoded by an integer matrix A, we associate a matroid structure called the bouquet graph of A and introduce another toric ideal called the bouquet ideal of A. We show how these objects capture the essential combinatorial and algebraic information about I A . Passing from the toric ideal to its bouquet ideal reveals a structure that allows us to classify several cases. For example, on the one end of the spectrum, there are ideals that we call stable, for which bouquets capture the complexity of various generating sets as well as the minimal free resolution. On the other end of the spectrum lie toric ideals whose various bases (e.g., minimal generating sets, Gröbner, Graver bases) coincide. Apart from allowing for classification-type results, bouquets provide a new way to construct families of examples of toric ideals with various interesting properties, such as robustness, genericity, and unimodularity. The new bouquet framework can be used to provide a characterization of toric ideals whose Graver basis, the universal Gröbner basis, any reduced Gröbner basis and any minimal generating set coincide.2010 Mathematics Subject Classification. 14M25, 13P10, 05C65, 13D02.

Abstract. Let V be an affine toric variety of codimension r over a field of any characteristic. We completely characterize the affine toric varieties that are set-theoretic complete intersections on binomials. In particular we prove that in the characteristic zero case, V is a set-theoretic complete intersection on binomials if and only if V is a complete intersection. Moreover, if F 1 , . . . , Fr are binomials such that . . . , Fr). While in the positive characteristic p case, V is a set-theoretic complete intersection on binomials if and only if V is completely p-glued.These results improve and complete all known results on these topics.