On the toric ideal of a matroid

TL;DR

This paper proves White's conjecture about the generators of the toric ideal of a matroid, showing they are equal up to saturation, and confirms the conjecture for strongly base orderable matroids.

Contribution

It establishes the equality of the toric ideal and the ideal generated by quadratic binomials up to saturation, advancing understanding of matroid-related algebraic structures.

Findings

Proved White's conjecture up to saturation.

Confirmed the conjecture for strongly base orderable matroids.

Both ideals define the same projective scheme.

Abstract

Describing minimal generating set of a toric ideal is a well-studied and difficult problem. In 1980 White conjectured that the toric ideal associated to a matroid is equal to the ideal generated by quadratic binomials corresponding to symmetric exchanges. We prove White's conjecture up to saturation, that is that the saturations of both ideals are equal. In the language of algebraic geometry this means that both ideals define the same projective scheme. Additionally we prove the full conjecture for strongly base orderable matroids.