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Roots of Ehrhart polynomials of Gorenstein Fano polytopes


Authors: Takayuki Hibi, Akihiro Higashitani and Hidefumi Ohsugi
Journal: Proc. Amer. Math. Soc. 139 (2011), 3727-3734
MSC (2010): Primary 52B20; Secondary 52B12
DOI: https://doi.org/10.1090/S0002-9939-2011-11013-X
Published electronically: March 30, 2011
MathSciNet review: 2813402
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Abstract: Given arbitrary integers $ k$ and $ d$ with $ 0 \leq 2k \leq d$, we construct a Gorenstein Fano polytope $ \mathcal{P} \subset \mathbb{R}^d$ of dimension $ d$ such that (i) its Ehrhart polynomial $ i(\mathcal{P}, n)$ possesses $ d$ distinct roots; (ii) $ i(\mathcal{P}, n)$ possesses exactly $ 2k$ non-real roots and $ d - 2k$ real roots; (iii) the real part of each of the non-real roots is equal to $ - 1 / 2$; (iv) all of the real roots belong to the open interval $ (-1, 0)$.


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Additional Information

Takayuki Hibi
Affiliation: Department of Pure and Applied Mathematics, Graduate School of Information Science and Technology, Osaka University, Toyonaka, Osaka 560-0043, Japan
Email: hibi@math.sci.osaka-u.ac.jp

Akihiro Higashitani
Affiliation: Department of Pure and Applied Mathematics, Graduate School of Information Science and Technology, Osaka University, Toyonaka, Osaka 560-0043, Japan
Email: sm5037ha@ecs.cmc.osaka-u.ac.jp

Hidefumi Ohsugi
Affiliation: Department of Mathematics, College of Science, Rikkyo University, Toshima-ku, Tokyo 171-8501, Japan
Email: ohsugi@rikkyo.ac.jp

DOI: https://doi.org/10.1090/S0002-9939-2011-11013-X
Keywords: Ehrhart polynomial, $\delta$-vector, Gorenstein Fano polytope.
Received by editor(s): September 3, 2010
Published electronically: March 30, 2011
Additional Notes: This research was supported by JST, CREST
Communicated by: Jim Haglund
Article copyright: © Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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