Ehrhart polynomials: inequalities and extremal constructions

This workshop was motivated by open questions in Ehrhart theory, a fundamental area of discrete geometry concerned with counting lattice points in polytopes and their dilates.

Let $P$ be a $d$-dimensional lattice polytope. For a nonnegative integer $t$ and polytope $P$, the $t$th dilate of $P$, denoted $tP$, is given by $tP:=\{tp : p \in P\}$. To $P$, we associate its Ehrhart function (or lattice point enumerator), denoted $i(P;t)$, which enumerates the lattice points contained in nonnegative integral dilates of $P$, i.e., $i(P;t) := |tP \cap \Z^d|$. Ehrhart showed that this function is a polynomial in $t$ of degree $d$. Moreover, due to Stanley, it is known that the generating function encoding this polynomial, the Ehrhart series of $P$, is a rational function $$\mathrm{Ehr}_{P}(z) := \sum_{t\geq 0} i(P;t)z^t = \frac{\sum_{j=0}^{d} h_{j}^* z^j}{(1-z)^{d+1}},$$ where $h_0^* = 1$ and $h_j^{*}\geq 0$ for all $j$. We call the numerator of the Ehrhart series, denoted $h^*(P;z)$, the $h^*$-polynomial of $P$, and the vector of its coefficients, $h^*(P) = (h_0^*,h_1^*,\ldots, h_d^*)$, the $h^*$-vector.

In this workshop, we investigated a number of questions concerning Ehrhart positivity and $h^*$ real-rootedness/unimodality for a wide range of classes of polytopes.