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Some open problems in low dimensional dynamical systems
The aim of this paper is to share with the mathematical community a list of
33 problems that I have found along the years during my research. I believe
that it is worth to think about them and, hopefully, it will be possible either
to solve some of the problems or to make some substantial progress. Many of
them are about planar differential equations but there are also questions about
other mathematical aspects: Abel differential equations, difference equations,
global asymptotic stability, geometrical questions, problems involving
polynomials or some recreational problems with a dynamical component
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
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