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

The local $h$-polynomial was introduced by Stanley as a fundamental enumerative invariant of a triangulation $\Delta$ of a simplex. This polynomial is known to have nonnegative and symmetric coefficients and is conjectured to be $\gamma$-positive when $\Delta$ is flag. This paper shows that the local $h$-polynomial has the stronger property of being real-rooted when $\Delta$ is the barycentric subdivision of an arbitrary geometric triangulation $\Gamma$ 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 $\Delta$, which is valid when $\Delta$ is any uniform triangulation of $\Gamma$. A combinatorial interpretation of the local $h$-polynomial of the second barycentric subdivision of the simplex is deduced.