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.
