White's conjecture for matroids and inner projections

TL;DR

This paper explores White's conjecture in matroid theory, demonstrating that its validity for one matroid implies its validity for related matroids differing by a basis, with motivation from algebraic variety projections.

Contribution

It establishes a reduction step for White's conjecture across matroids differing by a basis, linking combinatorial conjectures to algebraic geometry.

Findings

White's conjecture for one matroid implies it for matroids differing by one basis.

The study connects matroid theory with inner projections of algebraic varieties.

Provides a new approach to verify White's conjecture through basis modifications.

Abstract

White's conjecture predicts quadratic generators for the ideal of any matroid base polytope. We prove that White's conjecture for any matroid $M$ implies it also for any matroid $M'$, where $M$ and $M'$ differ by one basis. Our study is motivated by inner projections of algebraic varieties.