Minimal systems of binomial generators and the indispensable complex of a toric ideal

Abstract:

Let $A=\{\textbf {a}_1,\ldots ,\textbf {a}_m\} \subset \mathbb {Z}^n$ be a vector configuration and $I_A \subset K[x_1,\ldots ,x_m]$ 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 $\Delta _{\mathrm {ind}(A)}$. We show that the vertices of $\Delta _{\mathrm {ind}(A)}$ correspond to the indispensable monomials of the toric ideal $I_A$, while one dimensional facets of $\Delta _{\mathrm {ind}(A)}$ with minimal binomial $A$-degree correspond to the indispensable binomials of $I_{A}$.