Triangulations of simplicial complexes and theta polynomials

TL;DR

The paper introduces theta polynomials as a simpler analogue of local $h$-polynomials to study triangulations of simplicial complexes. It proves a locality-type formula expressing $h(\Delta',x)$ via theta polynomials and barycentric-subdivision polynomials, enabling the transfer of unimodality and $\gamma$-positivity from links to global triangulations. It establishes monotonicity of theta polynomials under triangulations with the interior vertex property, derives unimodality and $\gamma$-positivity results for $h$-polynomials of Cohen–Macaulay complexes, and applies the theory to antiprism triangulations to verify Gal's conjecture in new cases. It also discusses conjectures linking theta-positivity to the Link Conjecture and provides partial confirmations, including results for antiprism subdivisions and related subdivision families.

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.

Theorems & Definitions (46)

• Theorem 2.1
• Proposition 2.2: BW08
• Proposition 2.3
• Lemma 2.4
• proof
• proof : Proof of Proposition \ref{['prop:h-sd']}
• Definition 3.1
• Proposition 3.2: Sta93
• Example 3.3
• Theorem 3.4