Face Numbers of Uniform Triangulations of Simplicial Complexes

A triangulation of a simplicial complex $\Delta $ is said to be 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. Furthermore, 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.