On the toric ideals of matroids of fixed rank

In 1980 White conjectured that the toric ideal associated to a matroid is generated by quadratic binomials corresponding to symmetric exchanges. Herzog and Hibi go even further - they ask if the toric ideal of a matroid possesses a Grobner basis of degree 2. We study these problems for a class of matroids of fixed rank, and obtain several finiteness results.

We prove White's conjecture for `high degrees'. That is, we prove that for all matroids of fixed rank r, homogeneous parts of degree at least c(r) of the corresponding toric ideals are generated by quadratic binomials corresponding to symmetric exchanges. This extends our previous result (with Mateusz Michalek) confirming the conjecture `up to saturation'. We also prove that for the class of matroids of fixed rank, there exists a common upper bound on the degree of a Grobner basis. Namely, we prove that the toric ideal of a matroid of rank r possesses a Grobner basis of degree at most 2(r + 3)!.