On the real-rootedness of the local $h$-polynomials of edgewise subdivisions

TL;DR

This paper proves Athanasiadis' conjecture that the local h-polynomials of edgewise subdivisions of simplices have only real roots, using interlacing polynomial theory to establish the result.

Contribution

It introduces a novel application of interlacing polynomial theory to confirm the real-rootedness of local h-polynomials in edgewise subdivisions.

Findings

Confirmed Athanasiadis' conjecture for all positive integers r

Established interlacing properties of related polynomials

Proved real-rootedness of local h-polynomials in this context

Abstract

Athanasiadis conjectured that, for every positive integer $r$, the local $h$-polynomial of the $r$th edgewise subdivision of any simplex has only real zeros. In this paper, based on the theory of interlacing polynomials, we prove that a family of polynomials related to the desired local $h$-polynomial is interlacing and hence confirm Athanasiadis' conjecture.