  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 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.
