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arXiv · 2003.13372

Face numbers of uniform triangulations of simplicial complexes

Christos A. Athanasiadis
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Abstract

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.

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Christos A. Athanasiadis. 2020-03-30. Face numbers of uniform triangulations of simplicial complexes. https://arxiv.org/abs/2003.13372

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