Local h-polynomials, uniform triangulations and real-rootedness

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 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.