36 research outputs found
Smooth Fano polytopes whose Ehrhart polynomial has a root with large real part
The symmetric edge polytopes of odd cycles (del Pezzo polytopes) are known as
smooth Fano polytopes. In this paper, 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.Comment: 4 pages, We changed the order of the auhors and omitted a lot of
parts of the paper. (If you are interested in omitted parts, then please read
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Smooth Fano polytopes whose Ehrhart polynomial has a root with large real part
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
On positivity of Ehrhart polynomials
Ehrhart discovered that the function that counts the number of lattice points
in dilations of an integral polytope is a polynomial. We call the coefficients
of this polynomial Ehrhart coefficients, and say a polytope is Ehrhart positive
if all Ehrhart coefficients are positive (which is not true for all integral
polytopes). The main purpose of this article is to survey interesting families
of polytopes that are known to be Ehrhart positive and discuss the reasons from
which their Ehrhart positivity follows. We also include examples of polytopes
that have negative Ehrhart coefficients and polytopes that are conjectured to
be Ehrhart positive, as well as pose a few relevant questions.Comment: 40 pages, 7 figures. To appear in in Recent Trends in Algebraic
Combinatorics, a volume of the Association for Women in Mathematics Series,
Springer International Publishin
Equivariant Ehrhart theory
Motivated by representation theory and geometry, we introduce and develop an
equivariant generalization of Ehrhart theory, the study of lattice points in
dilations of lattice polytopes. We prove representation-theoretic analogues of
numerous classical results, and give applications to the Ehrhart theory of
rational polytopes and centrally symmetric polytopes. We also recover a
character formula of Procesi, Dolgachev, Lunts and Stembridge for the action of
a Weyl group on the cohomology of a toric variety associated to a root system.Comment: 40 pages. Final version. To appear in Adv. Mat