White's conjecture for matroids and inner projections
Kangjin Han, Mateusz Michałek, Julian Weigert
TL;DR
White's conjecture predicts quadratic (weak form) or explicit binomial generators for the toric ideal $I_M$ of a matroid $M$, and its behavior under basis-modification is open. The authors prove that, whenever both $M$ and the modified matroid $M_b$ are matroids, the strong White's conjecture holds for $M$ if and only if it holds for $M_b$, using a detailed analysis of symmetric basis exchanges and elimination of type-$b$ cubic relations; algebraically this corresponds to the inner-projection relation $I_{M_b}=I_M\cap k[\{y_{b'}\}\setminus y_b]$. As a corollary, sparse paving matroids satisfy the strong conjecture (Bonin), and the work provides a geometric perspective linking base polytopes, toric ideals, and inner projections of toric varieties defined by matroids. The results offer a structural, inductive approach to White's conjecture and connect combinatorial modifications to algebraic-geometric operations, broadening the classes of matroids for which the conjecture is understood.
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.