Motivated by connections to intersection homology of toric morphisms, the motivic monodromy conjecture, and a question of Stanley, we study the structure of geometric 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 geometric 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.

Revised: 2020-08-19

Accepted: 2020-08-20

Published online: 2020-12-04

Classification: 05E45

Keywords: local $h$-polynomials, triangulations of simplices, geometric triangulations

@article{ALCO_2020__3_6_1417_0, author = {de Moura, Andr\'e and Gunther, Elijah and Payne, Sam and Schuchardt, Jason and Stapledon, Alan}, title = {Triangulations of simplices with vanishing local <span class="mathjax-formula">$h$</span>-polynomial}, journal = {Algebraic Combinatorics}, pages = {1417--1430}, publisher = {MathOA foundation}, volume = {3}, number = {6}, year = {2020}, doi = {10.5802/alco.146}, language = {en}, url = {alco.centre-mersenne.org/item/ALCO_2020__3_6_1417_0/} }

`de Moura, André; Gunther, Elijah; Payne, Sam; Schuchardt, Jason; Stapledon, Alan. Triangulations of simplices with vanishing local $h$-polynomial. Algebraic Combinatorics, Volume 3 (2020) no. 6, pp. 1417-1430. doi : 10.5802/alco.146. https://alco.centre-mersenne.org/item/ALCO_2020__3_6_1417_0/`

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