Geodesic
Smooth Fano polytopes whose Ehrhart polynomial has a root with large real part
Discrete mathematics & theoretical computer science, DMTCS Proceedings vol. AR, 24th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2012), DMTCS Proceedings vol. AR, 24th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2012) (2012).

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The symmetric edge polytopes of odd cycles (del Pezzo polytopes) are known as smooth Fano polytopes. In this extended abstract, we show that if the length of the cycle is 127, then the Ehrhart polynomial has a root whose real part is greater than the dimension. As a result, we have a smooth Fano polytope that is a counterexample to the two conjectures on the roots of Ehrhart polynomials. 
@article{DMTCS_2012_special_263_a27,
     author = {Ohsugi, Hidefumi and Shibata, Kazuki},
     title = {Smooth {Fano} polytopes whose {Ehrhart} polynomial has a root with large real part},
     journal = {Discrete mathematics & theoretical computer science},
     publisher = {mathdoc},
     volume = {DMTCS Proceedings vol. AR, 24th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2012)},
     year = {2012},
     doi = {10.46298/dmtcs.3041},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.46298/dmtcs.3041/}
}
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Ohsugi, Hidefumi; Shibata, Kazuki. Smooth Fano polytopes whose Ehrhart polynomial has a root with large real part. Discrete mathematics & theoretical computer science, DMTCS Proceedings vol. AR, 24th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2012), DMTCS Proceedings vol. AR, 24th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2012) (2012). doi : 10.46298/dmtcs.3041. http://geodesic.mathdoc.fr/articles/10.46298/dmtcs.3041/

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