Simplex faces and quadratic toric ideals of lattice polytopes

Abstract

We study interactions between simplex faces of lattice polytopes and quadratic generation of toric ideals. We prove that, under a mild condition on edges, if the toric ideal of a lattice polytope is generated by quadratic binomials, then every clique of its 1-skeleton is the vertex set of a face. In particular, if the toric ideal of a $(0,1)$-polytope is generated by quadratic binomials, then every clique of its 1-skeleton is the vertex set of a face. For $(0,1)$-polytopes satisfying condition (E), we characterize this clique-face property in terms of divisibility by quadratic monomials appearing in quadratic binomials of the toric ideal; as a consequence, such toric ideals have no indispensable monomials of degree $\ge 3$. We apply these results to edge polytopes and cut polytopes, for which the clique-face property is equivalent to quadratic generation. Finally, motivated by conjectures on quadratic toric ideals, we verify the clique-face property for simple polytopes, matroid independence polytopes, and matroid base polytopes, and discuss stable set polytopes.