Smooth Fano polytopes whose Ehrhart polynomial has a root with large real part

Hidefumi Ohsugi (1) , Kazuki Shibata (1)

1 Department of Mathematics [Rikkyo]

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.

Les polytopes d'arêtes symètriques de cycles impaires (del Pezzo polytopes) sont connus sous le nom de polytopes de Fano lisses. Dans ce rèsumè ètendu, nous montrons que si la longueur du cycle est 127, alors le polynôme d'Ehrhart a une racine dont la partie rèele est plus grande que la dimension. En consèquence, nous avons un polytope de Fano lisse qui est un contre exemple à deux conjectures sur les racines de polynômes d'Ehrhart.

Mots clés: Ehrhart polynomials; Grôbner bases; Gorenstein Fano polytopes.

Domaines: Mathématique discrète [cs.DM]

Hidefumi Ohsugi, Kazuki Shibata. Smooth Fano polytopes whose Ehrhart polynomial has a root with large real part. 24th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2012), 2012, Nagoya, Japan. pp.307-312, ⟨10.46298/dmtcs.3041⟩. ⟨hal-01283170⟩