Eulerian posets and $Z$-polynomials

TL;DR

This paper establishes a fundamental interpretation of $Z$-polynomials for Eulerian posets, linking them to toric $h$-polynomials and intersection cohomology, thus unifying various mathematical frameworks.

Contribution

It proves that the $Z$-polynomial of any Eulerian poset equals the toric $h$-polynomial of its interval poset, resolving a key open problem.

Findings

$Z$-polynomial coincides with the toric $h$-polynomial for Eulerian posets

Under polyhedral conditions, $Z$-polynomial relates to intersection cohomology Poincaré polynomial

Results connect Chow polynomials with Veronese transforms on polynomials

Abstract

Let $P$ be a finite partially ordered set. In a recent series of works, Proudfoot introduced the notion of $Z$-polynomials associated with $P$-kernels, providing a unified framework for various intersection cohomology Poincar\'e polynomials arising in diverse areas of mathematics. One of the problems posed by Proudfoot was to interpret the $Z$-polynomial in a fundamental setting -- namely, when $P$ is the lattice of faces of a convex polytope (or, more generally, an Eulerian poset). We resolve this problem by proving that the $Z$-polynomial of any Eulerian poset coincides with the toric $h$-polynomial of the poset of all (possibly empty) closed intervals of $P$, ordered by reverse inclusion. Under suitable polyhedral conditions, this result identifies the $Z$-polynomial of a polytope with the Poincar\'e polynomial of the intersection cohomology of an associated auxiliary polytope. We prove some results about the Chow polynomials of the poset of intervals of an Eulerian poset and relate them with the Veronese transforms on polynomials.