The Local $h$-Polynomials of Cluster Subdivisions Have Only Real Zeros

TL;DR

This paper proves that local h-polynomials of cluster subdivisions across all Cartan--Killing types have only real zeros, confirming a conjecture for type A and extending to others using advanced polynomial techniques.

Contribution

It establishes the real-rootedness of local h-polynomials for all Cartan--Killing types, confirming a conjecture for type A and generalizing to others.

Findings

Local h-polynomials are real-rooted for all types.

The proof uses multiplier sequences and Chebyshev polynomials.

Confirms conjecture for type A and extends to all types.

Abstract

Athanasiadis raised the question whether the local $h$-polynomials of type $A$ cluster subdivisions have only real zeros. In this paper, we confirm this conjecture and prove the real-rootedness of local $h$-polynomials for all the other Cartan--Killing types. Our proofs mainly involve multiplier sequences and Chebyshev polynomials of the second kind.