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ARITHMETIC ASPECTS OF SYMMETRIC EDGE POLYTOPES

Part of: Combinatorics Polytopes and polyhedra Computational aspects and applications Polynomials, rational functions

Published online by Cambridge University Press:  14 May 2019

Akihiro Higashitani ,
Katharina Jochemko  and
Mateusz Michałek
Akihiro Higashitani
Affiliation:
Department of Mathematics, Graduate School of Science, Kyoto Sangyo University, Kamigamo Motoyama, Kita-ku, Kyoto, 603-8555, Japan email ahigashi@cc.kyoto-su.ac.jp
Katharina Jochemko
Affiliation:
Department of Mathematics, Royal Institute of Technology (KTH), SE-100 44 Stockholm, Sweden email jochemko@kth.se
Mateusz Michałek
Affiliation:
Max Planck Institute for Mathematics in the Sciences, Inselstr. 22, 04103 Leipzig, Germany email Mateusz.Michalek@mis.mpg.de Institute of Mathematics of the Polish Academy of Sciences, ul. Śniadeckich 8, 00-656 Warszawa, Poland
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Abstract

We investigate arithmetic, geometric and combinatorial properties of symmetric edge polytopes. We give a complete combinatorial description of their facets. By combining Gröbner basis techniques, half-open decompositions and methods for interlacing polynomials we provide an explicit formula for the $h^{\ast }$-polynomial in case of complete bipartite graphs. In particular, we show that the $h^{\ast }$-polynomial is $\unicode[STIX]{x1D6FE}$-positive and real-rooted. This proves Gal’s conjecture for arbitrary flag unimodular triangulations in this case, and, beyond that, we prove a strengthening due to Nevo and Petersen [On $\unicode[STIX]{x1D6FE}$-vectors satisfying the Kruskal–Katona inequalities. Discrete Comput. Geom.45(3) (2011), 503–521].

MSC classification

Primary: 05A15: Exact enumeration problems, generating functions 52B12: Special polytopes (linear programming, centrally symmetric, etc.)
Secondary: 13P10: Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases) 26C10: Polynomials 52B15: Symmetry properties of polytopes 52B20: Lattice polytopes (including relations with commutative algebra and algebraic geometry)
Type
Research Article
Information
Mathematika , Volume 65 , Issue 3 , 2019 , pp. 763 - 784
Copyright
Copyright © University College London 2019 

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