\(q\)-analogues of Ehrhart polynomials (Q2806119)

To a lattice polytope \(P\subset\mathbb R^d\) (i.e., the convex hull of finitely many points in in \(\mathbb Z^d\)), we associate the integer-point counting function \(L_P(t):=\# \left( tP \cap\mathbb Z^d \right)\), defined for positive integers \(t\). \textit{E. Ehrhart}'s famous theorem [C. R. Acad. Sci., Paris 254, 616--618 (1962; Zbl 0100.27601)] asserts that \(L_P\) is a polynomial in \(t\); alternatively, its generating function is rational of the form NEWLINE\[NEWLINE\mathrm{Ehr}_P(z) := 1+\sum_{t\geq 1} L_P(t) z^t =\frac{h^*(z)}{(1-z)^{\dim(P)+1}}\eqno{(\star)}NEWLINE\]NEWLINE for some polynomial \(h^*(z)\) of degree \(\leq\dim(P)\). It is natural to extend the definition of \(L_P\) to a weighted sum over monomials with powers a linear form \(\lambda\) evaluated at the integer lattice, with accompanying generating function NEWLINE\[NEWLINE \mathrm{Ehr}_{P, \lambda}(q, z):=1+\sum_{t\geq 1}\sum_{m \in tP}q^{\lambda(m)}\, z^t. NEWLINE\]NEWLINE While these types of weighted sums have been considered for many years, the paper under review gives the first fundamental study of \(\mathrm{Ehr}_{P, \lambda} (q, z)\) from the viewpoint of \(q\)-series and \(q\)-analogues of integers. Starting with the \(q\)-analogue of \((\star)\), where the denominator has now (generally more than \(\dim(P) + 1\)) factors of the form \((1-q^jz)\), the main results include the novel fact that the \(q\)-analogue of \(L_P\) is a polynomial in \([t]_q = 1 + q + \cdots + q^{ t-1 }\), and a \(q\)-analogue of the Ehrhart-Macdonald reciprocity theorem, which gives the interior integer-point count through an evaluation of \(L_P\) at negative integers [\textit{I. G. Macdonald}, J. Lond. Math. Soc., II. Ser. 4, 181--192 (1971; Zbl 0216.45205)].NEWLINENEWLINEMotivation and applications include \(q\)-Bernoulli numbers [\textit{L. Carlitz}, Duke Math. J. 15, 987--1000 (1948; Zbl 0032.00304)], order polynomials [\textit{R. P. Stanley}, Discrete Comput. Geom. 1, 9--23 (1986; Zbl 0595.52008)], and \(P\)-partitions [\textit{R. P. Stanley}, Mem. Am. Math. Soc. 119, 104 p. (1972; Zbl 0246.05007)].