  A triangulation of a simplicial complex $\Delta$ is called uniform if the
$f$-vector of its restriction to a face of $\Delta$ depends only on the
dimension of that face. This paper proves that the entries of the $h$-vector of
a uniform triangulation of $\Delta$ can be expressed as nonnegative integer
linear combinations of those of the $h$-vector of $\Delta$, where the
coefficients depend only on the dimension of $\Delta$ and the $f$-vectors of
the restrictions of the triangulation to simplices of various dimensions.
Moreover, it provides information about these coefficients, including formulas,
recurrence relations and various interpretations, and gives a criterion for the
$h$-polynomial of a uniform triangulation to be real-rooted. These results
unify and generalize several results in the literature about special types of
triangulations, such as barycentric, edgewise and interval subdivisions.
