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 $Δ$ of a simplex. This polynomial is known to have nonnegative and symmetric coefficients and is conjectured to be $γ$-positive when $Δ$ is flag. This paper shows that the local $h$-polynomial has the stronger property of being real-rooted when $Δ$ is the barycentric subdivision of an arbitrary geometric triangulation $Γ$ 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 $Δ$, which is valid when $Δ$ is any uniform triangulation of $Γ$. A combinatorial interpretation of the local $h$-polynomial of the second barycentric subdivision of the simplex is deduced.