Browse issues
no. 6
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, Volume 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.

Received:
Revised:
Accepted:
Published online:
DOI: 10.5802/alco.146
Classification: 05E45
Keywords: 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
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights 
@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 $h$-polynomial},
     journal = {Algebraic Combinatorics},
     pages = {1417--1430},
     publisher = {MathOA foundation},
     volume = {3},
     number = {6},
     year = {2020},
     doi = {10.5802/alco.146},
     language = {en},
     url = {https://alco.centre-mersenne.org/articles/10.5802/alco.146/}
}
TY  - JOUR
AU  - de Moura, André
AU  - Gunther, Elijah
AU  - Payne, Sam
AU  - Schuchardt, Jason
AU  - Stapledon, Alan
TI  - Triangulations of simplices with vanishing local $h$-polynomial
JO  - Algebraic Combinatorics
PY  - 2020
SP  - 1417
EP  - 1430
VL  - 3
IS  - 6
PB  - MathOA foundation
UR  - https://alco.centre-mersenne.org/articles/10.5802/alco.146/
DO  - 10.5802/alco.146
LA  - en
ID  - ALCO_2020__3_6_1417_0
ER  - 
%0 Journal Article
%A de Moura, André
%A Gunther, Elijah
%A Payne, Sam
%A Schuchardt, Jason
%A Stapledon, Alan
%T Triangulations of simplices with vanishing local $h$-polynomial
%J Algebraic Combinatorics
%D 2020
%P 1417-1430
%V 3
%N 6
%I MathOA foundation
%U https://alco.centre-mersenne.org/articles/10.5802/alco.146/
%R 10.5802/alco.146
%G en
%F 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/articles/10.5802/alco.146/

[1] Athanasiadis, Christos A. A survey of subdivisions and local h-vectors, The mathematical legacy of Richard P. Stanley, Amer. Math. Soc., Providence, RI, 2016, pp. 39-51 | DOI | Zbl

[2] Athanasiadis, Christos A.; Savvidou, Christina The local h-vector of the cluster subdivision of a simplex, Sém. Lothar. Combin., Volume 66 (2011/12), Paper no. Art. B66c, 21 pages | MR | Zbl

[3] de Cataldo, Mark A.; Migliorini, Luca; Mustaţă, Mircea Combinatorics and topology of proper toric maps, J. Reine Angew. Math., Volume 744 (2018), pp. 133-163 | DOI | MR | Zbl

[4] Denef, Jan; Loeser, François Motivic Igusa zeta functions, J. Algebraic Geom., Volume 7 (1998) no. 3, pp. 505-537 | MR | Zbl

[5] Igusa, Jun-ichi Complex powers and asymptotic expansions. II. Asymptotic expansions, J. Reine Angew. Math., Volume 278/279 (1975), pp. 307-321 | DOI | MR | Zbl

[6] Katz, Eric; Stapledon, Alan Local h-polynomials, invariants of subdivisions, and mixed Ehrhart theory, Adv. Math., Volume 286 (2016), pp. 181-239 | DOI | MR | Zbl

[7] Stanley, Richard P. Subdivisions and local h-vectors, J. Amer. Math. Soc., Volume 5 (1992) no. 4, pp. 805-851 | DOI | MR | Zbl

[8] Stapledon, Alan Formulas for monodromy, Res. Math. Sci., Volume 4 (2017), Paper no. 8, 42 pages | DOI | MR | Zbl

Cited by Sources: