Chow polynomials of rank-uniform labeled posets

TL;DR

This paper introduces UMEL-shellable posets, a broad class including many classical geometries, and proves real-rootedness of their associated Chow and chain polynomials, advancing conjectures in combinatorics and algebraic geometry.

Contribution

It develops the theory of UMEL-shellable posets and establishes real-rootedness of key polynomials, unifying various combinatorial and geometric structures.

Findings

Proves real-rootedness of Chow polynomials for UMEL-shellable posets.

Unifies classical geometries within a common framework.

Links polynomial properties to algebro-geometric structures in matroid theory.

Abstract

We introduce and develop the theory of UMEL-shellable posets. These are posets equipped with an edge-lexicographical labeling satisfying certain uniformity and monotonicity properties. This framework encompasses classical families of combinatorial geometries, including uniform matroids, projective and affine geometries, braid matroids of type A and B, and all Dowling geometries. It also comprises all rank-uniform supersolvable lattices, and therefore also all rank-uniform distributive lattices. Our main result establishes real-rootedness phenomena for the Chow polynomials, the augmented Chow polynomials, and the chain polynomials associated with those posets, thus making simultaneous progress towards conjectures by Ferroni--Schr\"oter, Huh--Stevens, and Athanasiadis--Kalampogia-Evangelinou. In the special case of lattices of flats of matroids, the (augmented) Chow polynomials coincide with the Hilbert--Poincar\'e series of the Chow ring associated to the smooth and generally noncompact toric varieties of the (augmented) Bergman fan of the matroid, whereas the chain polynomial encodes the Hilbert--Poincar\'e series of the Stanley--Reisner ring of the Bergman complex of the matroid. Therefore, these real-rootedness results are tightly linked to the study of these algebro-geometric structures in matroid theory.