Triangulations of simplices with vanishing local h-polynomial

André De Moura,Alan Stapledon,Elijah Gunther,Sam Payne,Jason Schuchardt

doi:10.5802/alco.146

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.

Highlights

Let Γ be a triangulation of a simplex ∆ of dimension d−1
In dimension 3, any triangulation with vanishing local h-polynomial is obtained from the trivial subdivision by a sequence of conical facet refinements
Let Γ be a facet refinement of Γ along G, and suppose that ΓG is the join of two triangulations of faces of G, one of which has vanishing local h-polynomial

Summary

Introduction

Let Γ be a triangulation of a simplex ∆ of dimension d−1. The h-polynomial h(Γ; x) = h0 + h1x + · · · + hdxd is a common and convenient way of encoding the number of faces of Γ in each dimension. In dimension 2, any triangulation with vanishing local h-polynomial is obtained from either the trivial subdivision or the triforce by a sequence of conical facet refinements. For an iterated conical facet refinement of the trivial subdivision in dimension 2, the internal edge graph is a tree that contains exactly one of the vertices of the original triangle. In dimension 3, any triangulation with vanishing local h-polynomial is obtained from the trivial subdivision by a sequence of conical facet refinements. The proof again relies on an analysis of the internal edge graph, which in dimension three is a union of trees, each of which contains exactly one vertex supported on a face of codimension at least 2 in the original simplex. Purely combinatorial viewpoint; Stanley asked for a nice characterization of such triangulations in his original paper [7, Problem 4.13]

Preliminaries

The internal edge graph

Dimensions 2 and 3

Higher dimensions