Transactions of the American Mathematical Society (Trans. Amer. Math. Soc.) (20190101), 371, no.~3, 2021-2042. ISSN: 0002-9947 (print).eISSN: 1088-6850.
The Ehrhart polynomial of a $d$-dimensional lattice polytope $P$ is the polynomial $\roman{ehr}_P(\cdot)$ of degree $d$, which, when evaluated at positive integer values $k \in \Bbb{N}$, yields the number of lattice points in the $k$th dilate of $P$, that is, $\roman{ehr}_P(k) = |kP \cap \Bbb{Z}^d|$. The $h^*$-polynomial of $P$ is the numerator polynomial $h^*_P(t) = h_0 + h_1 t + \cdots + h_d t^d$ of the generating function of $\roman{ehr}_P(\cdot)$. \par The extensive study of Ehrhart polynomials over the last half-century has led to a variety of important and insightful results. However, fundamental questions on combinatorial interpretations of the $h_i$ and classifications of the family of all $h^*$-polynomials are still wide open. A more specific example is the question of unimodality of the sequence $h_0,h_1,\ldots,h_d$ for certain families of lattice polytopes, which relates to the integer decomposition property, and thus to commutative algebra and integer programming. \par The paper at hand investigates the above questions on the class of lattice zonotopes, that is, polytopes of the form $Z = \{ \sum_{i=1}^n \lambda_i v_i : 0 \leq \lambda_i \leq 1 \}$, for some generators $v_1,\ldots,v_n \in \Bbb{Z}^d$. The coefficients of the Ehrhart polynomial $\roman{ehr}_Z(\cdot)$ of zonotopes have a beautiful combinatorial meaning, explored by Stanley (1991). Moreover, zonotopes constitute a broad class of polytopes relevant in many areas of mathematics. \par The first main result of the paper (Theorem 1.2) is that the $h^*$-polynomial of a lattice zonotope has only real roots and that its coefficients yield a unimodal sequence with peak in the middle. The authors prove their result in a much more general setting, when the counting function $P \mapsto |P \cap \Bbb{Z}^d|$ is replaced by any combinatorially positive valuation in the sense of [K. Jochemko and R. Sanyal, J. Eur. Math. Soc. (JEMS) {\bf 20} (2018), no.~9, 2181--2208; MR3836844]. Thus, this offers a vast extension of [J. Schepers and L. Van~Langenhoven, Ann. Comb. {\bf 17} (2013), no.~3, 571--589; MR3090179] where unimodality (without peak location) has been established for the $h^*$-polynomial of lattice parallelepipeds. \par The second main result (Theorem 1.3) is the description of the convex hull of the $h^*$-polynomials of all $d$-dimensional lattice zonotopes as a $d$-dimensional simplicial cone whose extreme rays correspond to certain polynomials $A_j(d+1,t)$, $1 \leq j \leq d+1$, that encode the distribution of refined descent statistics on permutations. \par Finally, the authors provide (in Theorem 1.4) a closed formula for the $h^*$-polynomial of a lattice zonotope $Z$, by identifying the coefficient in front of $A_j(d+1,t)$ as the sum of the number of relative interior lattice points in certain (lower-dimensional) parallelepipeds spanned by the generators of $Z$. \par The principal technique is to investigate half-open lattice parallelepipeds, prove the results on this class, and then carefully extend to lattice zonotopes via half-open decompositions.