Triangulations of simplices with vanishing local h-polynomial
de Moura, André1 ; Gunther, Elijah2 ; Payne, Sam3 ; Schuchardt, Jason4 ; Stapledon, Alan
1 Viela do Mato 4 BL A RC Esq Quinta da Beloura 2710-695 Sintra, Portugal
2 Departement of Mathematics David Rittenhouse Lab 209 South 33rd Street Philadelphia, PA 19104-6395, USA
3 UT Austin Departement of Mathematics 2515 Speedway, PMA 8.100 Austin TX 78722, USA
4 UCLA Departement of Mathematics Math. Sciences Building 6363 520 Portola Plaza Los Angeles, CA 90095, USA
Algebraic Combinatorics, Tome 3 (2020) no. 6, pp. 1417-1430.

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.

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DOI : 10.5802/alco.146
Classification : 05E45
Mots-clés : local $h$-polynomials, triangulations of simplices, geometric triangulations
de Moura, André 1 ; Gunther, Elijah 2 ; Payne, Sam 3 ; Schuchardt, Jason 4 ; Stapledon, Alan 

1 Viela do Mato 4 BL A RC Esq Quinta da Beloura 2710-695 Sintra, Portugal
2 Departement of Mathematics David Rittenhouse Lab 209 South 33rd Street Philadelphia, PA 19104-6395, USA
3 UT Austin Departement of Mathematics 2515 Speedway, PMA 8.100 Austin TX 78722, USA
4 UCLA Departement of Mathematics Math. Sciences Building 6363 520 Portola Plaza Los Angeles, CA 90095, USA 
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     title = {Triangulations of simplices with vanishing local $h$-polynomial},
     journal = {Algebraic Combinatorics},
     pages = {1417--1430},
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     volume = {3},
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de Moura, André; Gunther, Elijah; Payne, Sam; Schuchardt, Jason; Stapledon, Alan. Triangulations of simplices with vanishing local $h$-polynomial. Algebraic Combinatorics, Tome 3 (2020) no. 6, pp. 1417-1430. doi : 10.5802/alco.146. http://www.numdam.org/articles/10.5802/alco.146/

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