Triangulations of simplicial complexes and theta polynomials

TL;DR

This paper introduces theta polynomials as a simpler alternative to local h-polynomials for studying triangulations of simplicial complexes, enabling new proofs of properties like gamma-positivity and confirming Gal's conjecture in specific cases.

Contribution

It develops a parallel theory replacing local h-polynomials with theta polynomials, leading to new insights into the unimodality and gamma-positivity of h-polynomials.

Findings

Theta polynomials are simpler to analyze than local h-polynomials.

Gamma-positivity of h-polynomials is established for antiprism triangulations of homology spheres.

Confirms Gal's conjecture for a new class of simplicial complexes.

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.