Nonnegativity of the $g$-polynomial of split matroids

Source abstract

We prove that the $g$-polynomial of every split matroid has nonnegative coefficients, establishing Speyer's conjecture for a class closed under taking minors and containing all paving and copaving matroids. Our proof uses a deletion--contraction identity obtained by constructing an auxiliary split matroid. We first show that every simple, cosimple, connected split matroid $M$ has an element $e$ for which both the deletion $M\setminus e$ and the contraction $M/e$ are connected. More generally, let $M$ be any connected split matroid of rank $k$ on a ground set $E$ with $|E|\ge4$. Suppose that $e\in E$ is such that both $M\setminus e$ and $M/e$ are connected. For every $v\in E\setminus\{e\}$, we construct a connected elementary split matroid $N_{e,v}$ of rank $k-1$ on $E\setminus\{e,v\}$ satisfying $$g_M(t)=g_{M\setminus e}(t)+g_{M/e}(t)+t\,g_{N_{e,v}}(t).$$ Using a fixed total order on $E$, we prescribe the proper cyclic flats of $N_{e,v}$ and their ranks. The identity follows from the covaluative formula of Ferroni and Schröter together with recurrences for its correction polynomials, derived from Ferroni's enumeration of admissible Delannoy paths for Schubert matroids. Since all three matroids on the right have fewer elements, the identity supplies the induction step in the proof of nonnegativity.