49 research outputs found
Examples and counterexamples in Ehrhart theory
This article provides a comprehensive exposition about inequalities that the
coefficients of Ehrhart polynomials and -polynomials satisfy under various
assumptions. We pay particular attention to the properties of Ehrhart
positivity as well as unimodality, log-concavity and real-rootedness for
-polynomials.
We survey inequalities that arise when the polytope has different normality
properties. We include statements previously unknown in the Ehrhart theory
setting, as well as some original contributions in this topic. We address
numerous variations of the conjecture asserting that IDP polytopes have a
unimodal -polynomial, and construct concrete examples that show that these
variations of the conjecture are false. Explicit emphasis is put on polytopes
arising within algebraic combinatorics.
Furthermore, we describe and construct polytopes having pathological
properties on their Ehrhart coefficients and roots, and we indicate for the
first time a connection between the notions of Ehrhart positivity and
-real-rootedness. We investigate the log-concavity of the sequence of
evaluations of an Ehrhart polynomial at the non-negative integers. We
conjecture that IDP polytopes have a log-concave Ehrhart series. Many
additional problems and challenges are proposed.Comment: Comments welcome
Ehrhart Positivity for Generalized Permutohedra
International audienceThere are few general results about the coefficients of Ehrhart polynomials. We present a conjecture about their positivity for a certain family of polytopes known as generalized permutohedra. We have verified the conjecture for small dimensions combining perturbation methods with a new valuation on the algebra of rational pointed polyhedral cones constructed by Berline and Vergne.Il existe peu de résultats sur les coefficients des polynômes d’Ehrhart. On présente une conjecture concernant leur positivité pour une certaine famille de polytopes connus sous le nom de permutoèdre généralisé. On a vérifié la conjecture pour les petites dimensions en combinant des méthodes de perturbation avec une nouvelle valuation sur l’algèbre des cônes polyédraux rationnels pointés, construite par Berline et Vergne
Polytopal and structural aspects of matroids and related objects
PhDThis thesis consists of three self-contained but related parts. The rst is focussed on
polymatroids, these being a natural generalisation of matroids. The Tutte polynomial is
one of the most important and well-known graph polynomials, and also features prominently
in matroid theory. It is however not directly applicable to polymatroids. For
instance, deletion-contraction properties do not hold. We construct a polynomial for
polymatroids which behaves similarly to the Tutte polynomial of a matroid, and in fact
contains the same information as the Tutte polynomial when we restrict to matroids.
The second section is concerned with split matroids, a class of matroids which arises by
putting conditions on the system of split hyperplanes of the matroid base polytope. We
describe these conditions in terms of structural properties of the matroid, and use this
to give an excluded minor characterisation of the class.
In the nal section, we investigate the structure of clutters. A clutter consists of a nite
set and a collection of pairwise incomparable subsets. Clutters are natural generalisations
of matroids, and they have similar operations of deletion and contraction. We introduce
a notion of connectivity for clutters that generalises that of connectivity for matroids.
We prove a splitter theorem for connected clutters that has the splitter theorem for
connected matroids as a special case: if M and N are connected clutters, and N is a
proper minor of M, then there is an element in E(M) that can be deleted or contracted
to produce a connected clutter with N as a minor