On chain polynomials of geometric lattices

TL;DR

This paper verifies the conjecture that chain polynomials of certain geometric lattices have only real zeros, including specific classes like Dowling lattices and paving matroids, and explores how this property behaves under various lattice operations.

Contribution

It confirms the real-zero property for several classes of geometric lattices and analyzes its stability under lattice operations.

Findings

Chain polynomials of these lattices have only real zeros.

The property is preserved under direct products, ordinal sums, and extensions.

The conjecture holds for all lattices of flats of paving matroids.

Abstract

Athanasiadis and Kalampogia-Evangelinou recently conjectured that the chain polynomial of any geometric lattice has only real zeros. We verify this conjecture for families of geometric lattices including perfect matroid designs, Dowling lattices, and for a class of geometric lattices that contains all lattices of flats of paving matroids. We also investigate how the conjecture behaves with respect to certain operations such as direct products, ordinal sums and single-element extensions.