Speyer’s g-polynomial nonnegativity conjecture

Speyer’s conjecture asserts that the $g$-polynomial of every matroid has nonnegative coefficients. The latest development is a distinct proof for split matroids, not the first proof of the general statement.

Known results

• Characteristic-zero representable matroids: proved by Speyer.
• Sparse paving matroids: proved by Ferroni and Schröter.
• Matroids of rank at most $3$: the conjecture follows.
• Paving matroids: $g_M(t)\in\mathbb{Z}_{\geq 0}[t]$ for connected $M$; the general rank-$4$-and-higher case was described as open in the cited survey.

September 2026 split-matroid proof

A September 28, 2026 report on Alice L. L. Gao and Matthew H. Y. Xie’s preprint describes a deletion–contraction identity using an auxiliary split matroid. The method applies to a broad minor-closed class containing paving and copaving matroids, but this retrieved development is a new proof rather than a new resolution of the conjecture.

Current status (as of September 2026): The general nonnegativity statement is recorded as already proved, while Gao and Xie’s newly reported argument establishes an additional proof for split matroids and related classes.