Triangulations of simplices with vanishing local h-polynomial
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: 2020-03-09
Revised: 2020-08-19
Accepted: 2020-08-20
Published online: 2020-12-04
DOI: https://doi.org/10.5802/alco.146
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/

[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 | Article | Zbl 1370.05225

[2] Athanasiadis, Christos A.; Savvidou, Christina The local h-vector of the cluster subdivision of a simplex, Sém. Lothar. Combin., Volume 66 (2011/12), Art. B66c, 21 pages | MR 2971012 | Zbl 1253.05150

[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 | Article | MR 3871442 | Zbl 1408.14159

[4] Denef, Jan; Loeser, François Motivic Igusa zeta functions, J. Algebraic Geom., Volume 7 (1998) no. 3, pp. 505-537 | MR 1618144 | Zbl 0943.14010

[5] Igusa, Jun-ichi Complex powers and asymptotic expansions. II. Asymptotic expansions, J. Reine Angew. Math., Volume 278/279 (1975), pp. 307-321 | Article | MR 404215 | Zbl 0315.41029

[6] Katz, Eric; Stapledon, Alan Local h-polynomials, invariants of subdivisions, and mixed Ehrhart theory, Adv. Math., Volume 286 (2016), pp. 181-239 | Article | MR 3415684 | Zbl 1325.05192

[7] Stanley, Richard P. Subdivisions and local h-vectors, J. Amer. Math. Soc., Volume 5 (1992) no. 4, pp. 805-851 | Article | MR 1157293 | Zbl 0768.05100

[8] Stapledon, Alan Formulas for monodromy, Res. Math. Sci., Volume 4 (2017), 8, 42 pages | Article | MR 3633400 | Zbl 1401.14049