Triangulations of simplicial complexes and theta polynomials

arXiv2209.01674MaRDI QIDQ6409709

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Publication date: 4 September 2022

Abstract: The theta polynomial of a triangulation $Delta$ of a ball with boundary $partialDelta$ is defined as the difference of the $h$-polynomial of $partialDelta$ from the $h$-polynomial of $Delta$. A basic theory for the face enumeration of triangulations of simplicial complexes, parallel to that of Stanley, is developed in which the role of local $h$-polynomials is played by theta polynomials. Positivity and monotonicity properties of theta polynomials are investigated and shown to have strong implications in the study of $h$-polynomials and local $h$-polynomials. In particular, some unimodality and gamma-positivity properties of barycentric subdivisions are extended to larger classes of triangulations, including antiprism triangulations.

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