Séminaire Lotharingien de Combinatoire, 95B.20 (2026), 12 pp.
Petter Brändén and Lorenzo Vecchi
Chow Polynomials of Totally Nonnegative Matrices and Posets
Abstract.
Huh-Stevens and Ferroni-Schröter independently conjectured that
Hilbert-Poincaré series of Chow rings of geometric lattices have only
real zeros, and Ferroni, Matherne and the second author extended this
conjecture to Chow polynomials of Cohen-Macaulay posets. In this paper
we address the above conjectures by providing new defining relations
and properties of Chow functions of posets and matrices.
These are used to prove that Chow polynomials of totally nonnegative
matrices have only real zeros, which proves the above conjectures for
projective and affine geometries, face lattices of cubical polytopes,
partition lattices and Dowling lattices, perfect matroid designs, and
lattices of flats of paving matroids.
We also study Chow polynomials of Toeplitz matrices in greater detail,
and show how these are related to the combinatorics of binomial and
Sheffer posets, as well as to a family of generalized Eulerian
polynomials that have been studied by e.g. Stanley, Brenti,
Stembridge and Shareshian-Wachs.
Received: November 15, 2025.
Accepted: February 15, 2026.
Final version: April 1, 2026.
The following versions are available: