Nonnegativity of the g-polynomial for all matroids

Determine whether the g-polynomial of every matroid has nonnegative integer coefficients, thereby resolving Speyer’s conjecture beyond the currently known cases of matroids of rank or corank at most three.

Background

Speyer conjectured that the g-polynomial of every matroid belongs to $\mathbb{Z}_{\ge 0}[t]$. Prior results established nonnegativity for realizable matroids over fields of characteristic zero, for the leading coefficient, and for matroids of rank at most three; duality gives the corresponding result for matroids of corank at most three. The paper proves the conjecture for split matroids, a minor- and duality-closed class containing all paving and copaving matroids, but explicitly notes that the general problem remains unresolved.