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arXiv · 2209.01674

Triangulations of simplicial complexes and theta polynomials

Christos A. Athanasiadis
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Abstract

An enumerative theory of triangulations of simplicial complexes has been developed by Stanley. A key role in his theory is played by the local $h$-polynomial of a triangulation of a simplex. This paper develops a parallel theory, in which the role of the local $h$-polynomial is played by a simpler invariant, namely the theta polynomial. This allows one to deduce unimodality and gamma-positivity properties of $h$-polynomials of triangulations of simplicial complexes from corresponding properties of theta polynomials, which are studied here in some detail. To mention one concrete application, the $h$-polynomial of the antiprism triangulation of any simplicial homology sphere is shown to be gamma-positive, thus confirming Gal's conjecture in a new special case.

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Christos A. Athanasiadis. 2022-09-04. Triangulations of simplicial complexes and theta polynomials. https://arxiv.org/abs/2209.01674

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