Some Computational Study of the Root Distribution of Ehrhart Polynomials

Masahiro Hachimori and Yumi Yamada
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Masahiro Hachimori: Faculty of Engineering, Information and Systems, University of Tsukuba, Tsukuba, Ibaraki 305-8573, Japan
Yumi Yamada: Department of Policy and Planning Sciences, Graduate School of Systems and Information Engineering, University of Tsukuba, Tsukuba, Ibaraki 305-8573, Japan

International Game Theory Review (IGTR), 2023, vol. 25, issue 03, 1-15

Abstract: In this paper, we investigate the root distribution of the Ehrhart polynomials of lattice polytopes. When the lattice polytope is reflexive, the roots of the Ehrhart polynomial distribute symmetrically with respect to the line Re(z) = −1 2. A special case of this distribution is when all the roots lie on this line. Our main concern is to find out which reflexive polytopes satisfy this special condition. Such lattice polytopes are called CL-polytopes. Another special case opposite to this is when all the roots are real. Such lattice polytopes are called real polytopes. The first topic of this paper is the Ehrhart polynomials of equatorial spheres, which are related to graded posets. We discuss the CL-ness of the equatorial Ehrhart polynomials for complete graded posets and zig-zag posets. The second topic is to investigate the relation of the root distribution of the Ehrhart polynomials of a reflexive polytope Q and its dual Q∨. We discuss the CL-ness and realness of Q and Q∨ in pair, of dimensions up to 4. Throughout this paper, we investigate these problems by computer calculation.

Keywords: Lattice polytope; Ehrhart polynomial; root distribution; poset (search for similar items in EconPapers)
Date: 2023
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DOI: 10.1142/S0219198923400054

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