Pure O-sequences: known results, applications, and open problems

This note presents a discussion of the algebraic and combinatorial aspects of the theory of pure O-sequences. Various instances where pure O-sequences appear are described. Several open problems that deserve further investigation are also presented.Comment: Some minor revisions with respect to the previous version. 20 pages. To appear in a Springer volume edited by Irena Peeva and dedicated to David Eisenbud on the occasion of his 65th birthda

Examples and counterexamples in Ehrhart theory

This article provides a comprehensive exposition about inequalities that the coefficients of Ehrhart polynomials and $h^*$-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 $h^*$-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 $h^*$-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 $h^*$-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

Unimodality Questions in Ehrhart Theory

An interesting open problem in Ehrhart theory is to classify those lattice polytopes having a unimodal h*-vector.Although various sufficient conditions have been found, necessary conditions remain a challenge. Highly-structured polytopes, such as the polytope of real doubly-stochastic matrices, have been proven to possess unimodal h*-vectors, but the same is unknown even for small variations of it. In this dissertation, we mainly consider two particular classes of polytopes: reflexive simplices and the polytope of symmetric real doubly-stochastic matrices. For the first class, we discuss an operation that preserves reflexivity, integral closure, and unimodality of the h*-vector, providing one explanation for why unimodality occurs in this setting. We also discuss the failure of proving unimodality in this setting using weak Lefschetz elements. With the second class, we prove partial unimodality results by examining their toric ideals and using a correspondence between these and regular triangulations of the polytopes. Lastly, we describe the computational methods used to help develop these results. Several software programs were used, and the code has proven useful outside of the main focus of this work

Log-concavity and compressed ideals in certain Macaulay posets

Let Bn be the poset of subsets of {1; 2; : : : ; n} ordered byinclusion and Mn be the poset of monomials in x1; x2; : : : ; xn ordered bydivisibility. Both these posets have an additional linear order making them what is called Macaulayposets. We show in this paper that the pro-les (also called f-vectors) of ideals in Bn and Mn generated bythe -rst elements (relativelyto the linear order) of a given rank are log-concave