Local $h$-polynomials, uniform triangulations and real-rootedness

Abstract

The local $h$-polynomial was introduced by Stanley as a fundamental enumerative invariant of a triangulation $\hat{I}”$ of a simplex. This polynomial is known to have nonnegative and symmetric coefficients and is conjectured to be ${\hat{I}}^3$-positive when $\hat{I}”$ is flag. This paper shows that the local $h$-polynomial has the stronger property of being real-rooted when $\hat{I}”$ is the barycentric subdivision of an arbitrary geometric triangulation $\hat{I}“$ of the simplex. An analogous result for edgewise subdivisions is proven. The proofs are based on a new combinatorial formula for the local $h$-polynomial of $\hat{I}”$, which is valid when $\hat{I}”$ is any uniform triangulation of $\hat{I}“$. A combinatorial interpretation of the local $h$-polynomial of the second barycentric subdivision of the simplex is deduced.

Final version