Face numbers of barycentric subdivisions of cubical complexes

TL;DR

This paper proves that the $h$-polynomial of barycentric subdivisions of certain cubical complexes has only real roots and is interlaced with type $B_n$ Eulerian polynomials, confirming a conjecture for shellable cubical complexes.

Contribution

It establishes the real-rootedness and interlacing properties of the $h$-polynomial for barycentric subdivisions of cubical complexes, extending known results to new classes.

Findings

$h$-polynomial has only real roots

Interlacing with type $B_n$ Eulerian polynomial

Applies to shellable cubical complexes and convex polytopes

Abstract

The $h$-polynomial of the barycentric subdivision of any $n$-dimensional cubical complex with nonnegative cubical $h$-vector is shown to have only real roots and to be interlaced by the Eulerian polynomial of type $B_n$. This result applies to barycentric subdivisions of shellable cubical complexes and, in particular, to barycentric subdivisions of cubical convex polytopes and answers affirmatively a question of Brenti, Mohammadi and Welker.