Compatible polynomials and edgewise subdivisions

Abstract

In this paper we answer a question posed by C. A. Athanasiadis. Namely, we prove that the local $h$-polynomial of the $r$th edgewise subdivision of the $(n-1)$-dimensional simplex $2^V$ has only real zeros. In doing so we find a tool, using compatible polynomials, that can be used to check if polynomials in certain types of families of polynomials have only real zeros.

This paper is part of my doctoral thesis. Corollary 2.5 of this paper was recently independently proved by P. B. Zhang