297 research outputs found
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
Interlacing Ehrhart Polynomials of Reflexive Polytopes
It was observed by Bump et al. that Ehrhart polynomials in a special family
exhibit properties similar to the Riemann {\zeta} function. The construction
was generalized by Matsui et al. to a larger family of reflexive polytopes
coming from graphs. We prove several conjectures confirming when such
polynomials have zeros on a certain line in the complex plane. Our main new
method is to prove a stronger property called interlacing
Roots of Ehrhart Polynomials of Smooth Fano Polytopes
V. Golyshev conjectured that for any smooth polytope P of dimension at most
five, the roots z\in\C of the Ehrhart polynomial for P have real part equal
to -1/2. An elementary proof is given, and in each dimension the roots are
described explicitly. We also present examples which demonstrate that this
result cannot be extended to dimension six.Comment: 10 page
Unimodality Problems in Ehrhart Theory
Ehrhart theory is the study of sequences recording the number of integer
points in non-negative integral dilates of rational polytopes. For a given
lattice polytope, this sequence is encoded in a finite vector called the
Ehrhart -vector. Ehrhart -vectors have connections to many areas of
mathematics, including commutative algebra and enumerative combinatorics. In
this survey we discuss what is known about unimodality for Ehrhart
-vectors and highlight open questions and problems.Comment: Published in Recent Trends in Combinatorics, Beveridge, A., et al.
(eds), Springer, 2016, pp 687-711, doi 10.1007/978-3-319-24298-9_27. This
version updated October 2017 to correct an error in the original versio
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