Triangulations of simplices with vanishing local h-polynomial

TL;DR

This paper investigates triangulations of a simplex with vanishing local $h$-polynomial, motivated by connections to intersection homology and motivic monodromy. The authors identify conical facet refinements that preserve the local $h$-polynomial and prove a complete classification in dimensions 2 and 3: such triangulations are generated from a small set of base subdivisions by iterative conical refinements. In higher dimensions, the internal edge graph analysis yields partial results but also shows that a finite generating set is unlikely, with explicit counterexamples. The results illuminate the combinatorial structure underlying vanishing local $h$-polynomials and suggest directions for connections to toric geometry and monodromy questions.

Abstract

Motivated by connections to intersection homology of toric morphisms, the motivic monodromy conjecture, and a question of Stanley, we study the structure of triangulations of simplices whose local h-polynomial vanishes. As a first step, we identify a class of refinements that preserve the local h-polynomial. In dimensions 2 and 3, we show that all triangulations with vanishing local h-polynomial are obtained from one or two simple examples by a sequence of such refinements. In higher dimensions, we prove some partial results and give further examples.

Figures (6)

• Figure 1: A facet refinement of the triforce along its lower left facet.
• Figure 2: A triangulation obtained from the triforce by a series of conical facet refinements.
• Figure 3: Possible locations for $v$ in Subcase 1.3
• Figure 4: Structure of $C$.
• Figure 5: A corner cutting edge $e$ with vertices $w_1$ and $w_2$

Theorems & Definitions (30)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Definition 2.1
• Definition 2.2
• Proposition 2.3
• proof
• Definition 2.4
• Corollary 2.5
• proof