On the toric ideals of matroids of a fixed rank

TL;DR

This paper proves White's conjecture for high degrees in the toric ideals of matroids of fixed rank, extending previous results and analyzing algebraic properties like Gr"obner bases and Betti tables.

Contribution

It confirms White's conjecture for large degrees and studies algebraic invariants of toric ideals of fixed-rank matroids.

Findings

White's conjecture holds for high degrees in fixed-rank matroids

Analysis of degrees of Gr"obner bases

Betti tables of toric ideals of matroids

Abstract

In $1980$ White conjectured that every element of the toric ideal of a matroid is generated by quadratic binomials corresponding to symmetric exchanges. We prove White's conjecture for high degrees with respect to the rank. This extends our result arXiv:1302.5236 confirming White's conjecture `up to saturation'. Furthermore, we study degrees of Gr\"{o}bner bases and Betti tables of the toric ideals of matroids of a fixed rank.