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.

中文翻译：

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单纯复形的均匀三角剖分的面数

一个单纯复形 $\Delta $ 的三角剖分被认为是一致的，如果它限制在 $\Delta $ 的面的 $f$-向量仅取决于那个面的维数。这篇论文证明了$\Delta$的均匀三角剖分的$h$-向量的入口可以表示为$\Delta$的$h$-向量的非负整数线性组合，其中系数仅取决于关于$\Delta $ 的维数和三角剖分限制到各种维数的单纯形的$f$-向量。此外，它提供了有关这些系数的信息，包括公式、递推关系和各种解释，并给出了统一三角剖分的 $h$-多项式为实根的标准。