Degree bounds for the toric ideal of a matroid

Michał Lasoń

Published 2016-01-29Version 1

Describing minimal generating set of a toric ideal, or the minimum degree in which it is generated, is a well-studied and difficult problem. In 1980 White conjectured that the toric ideal of a matroid is generated by quadratic binomials corresponding to symmetric exchanges. In particular, that it is generated in degree $2$. We prove that the toric ideal associated to a matroid of rank $r$ is generated in degree at most $(r+3)!$. As a corollary we obtain that checking if White's conjecture is true for matroids of a fixed rank is a decidable problem.