Tohoku Mathematical Journal
2026
September
SECOND SERIES VOL. 78, NO. 3

Tohoku Math. J.
78 (2026), 423-441

Title TRIANGULATIONS OF SIMPLICIAL COMPLEXES AND THETA POLYNOMIALS

Author Christos A. Athanasiadis

(Received January 6, 2025, revised March 3, 2025)
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.

Mathematics Subject Classification. Primary 05E45; Secondary 13C14, 55U10.
Key words and phrases. Triangulation, homology ball, theta polynomial, local $h$-polynomial, unimodality, $\gamma$-positivity.

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