The Gröbner Version of White's Conjecture is False

Abstract

We show that the toric ideal of the Fano matroid polytope does not have a quadratic Gröbner basis. This resolves in the negative a strong version of White's conjecture from matroid theory. This result was found independently by De Loera, Ferroni, Morales, and Rambau. Our approach is based on a new characterization of regular unimodular flag triangulations, which reduces the problem to an instance of SMT involving boolean and real variables. We then use an SMT solver to prove unsatisfiability. Using this approach, we also show that all 8-element matroids which do not have the Fano matroid or its dual as a minor, with the possible exception of the matroid $T_8$, have toric ideals which admit quadratic Gröbner bases.