ALGEBRAIC COMBINATORICS

no. 4
Toric ideals of Minkowski sums of unit simplices
Algebraic Combinatorics, Volume 3 (2020) no. 4, pp. 831-837.

In this paper, we discuss the toric ideals of the Minkowski sums of unit simplices. More precisely, we prove that the toric ideal of the Minkowski sum of unit simplices has a squarefree initial ideal and is generated by quadratic binomials. Moreover, we also prove that the Minkowski sums of unit simplices have the integer decomposition property. Those results are a partial contribution to Oda conjecture and Bøgvad conjecture.

Received: 2019-08-14
Revised: 2019-11-29
Accepted: 2020-03-02
Published online: 2020-08-20
DOI: https://doi.org/10.5802/alco.117
Classification: 13P10,  52B20
Keywords: Integer decomposition property, Gröbner basis, Generalized permutohedron. 
@article{ALCO_2020__3_4_831_0,
     author = {Higashitani, Akihiro and Ohsugi, Hidefumi},
     title = {Toric ideals of Minkowski sums of unit simplices},
     journal = {Algebraic Combinatorics},
     publisher = {MathOA foundation},
     volume = {3},
     number = {4},
     year = {2020},
     pages = {831-837},
     doi = {10.5802/alco.117},
     language = {en},
     url = {alco.centre-mersenne.org/item/ALCO_2020__3_4_831_0/}
}
Higashitani, Akihiro; Ohsugi, Hidefumi. Toric ideals of Minkowski sums of unit simplices. Algebraic Combinatorics, Volume 3 (2020) no. 4, pp. 831-837. doi : 10.5802/alco.117. https://alco.centre-mersenne.org/item/ALCO_2020__3_4_831_0/

[1] Aoki, Satoshi; Hibi, Takayuki; Ohsugi, Hidefumi; Takemura, Akimichi Gröbner bases of nested configurations, J. Algebra, Volume 320 (2008) no. 6, pp. 2583-2593 | Article | Zbl 1152.14047

[2] Beck, Matthias; Haase, Christian; Higashitani, Akihiro; Hofscheier, Johannes; Jochemko, Katharina; Katthän, Lukas; Michałek, Mateusz Smooth centrally symmetric polytopes in dimension 3 are IDP, Ann. Comb., Volume 23 (2019) no. 2, pp. 255-262 | Article | MR 3962856 | Zbl 1429.52015

[3] Bruns, Winfried The quest for counterexamples in toric geometry, Commutative algebra and algebraic geometry (CAAG-2010) (Ramanujan Math. Soc. Lect. Notes Ser.) Volume 17, Ramanujan Math. Soc., Mysore, 2013, pp. 45-61 | MR 3155951 | Zbl 1317.14112

[4] Buchstaber, Victor M.; Panov, Taras E. Toric topology, Mathematical Surveys and Monographs, Volume 204, American Mathematical Society, Providence, RI, 2015, xiv+518 pages | Article | MR 3363157 | Zbl 1375.14001

[5] Herzog, Jürgen; Hibi, Takayuki; Ohsugi, Hidefumi Binomial ideals, Graduate Texts in Mathematics, Volume 279, Springer, Cham, 2018, xix+321 pages | Article | MR 3838370 | Zbl 1403.13004

[6] Postnikov, Alexander Permutohedra, associahedra, and beyond, Int. Math. Res. Not. IMRN (2009) no. 6, pp. 1026-1106 | Article | MR 2487491 | Zbl 1162.52007

[7] Shibuta, Takafumi Gröbner bases of contraction ideals, J. Algebraic Combin., Volume 36 (2012) no. 1, pp. 1-19 | Article | MR 2927653 | Zbl 1250.13028

[8] Stanley, Richard P.; Pitman, Jim A polytope related to empirical distributions, plane trees, parking functions, and the associahedron, Discrete Comput. Geom., Volume 27 (2002) no. 4, pp. 603-634 | Article | MR 1902680 | Zbl 1012.52019

[9] Sturmfels, Bernd Gröbner bases and convex polytopes, University Lecture Series, Volume 8, American Mathematical Society, Providence, RI, 1996, xii+162 pages | Zbl 0856.13020

[10] Tsuchiya, Akiyoshi Cayley sums and Minkowski sums of 2-convex-normal lattice polytopes (2018) (https://arxiv.org/abs/1804.10538)