If \(\mathcal {P}\) is a lattice polytope (i.e., \(\mathcal {P}\) is the convex hull of finitely many integer points in \(\mathbb {R}^d\) ), Ehrhart’s famous theorem (1962) asserts that the integer-point counting function \(|t \mathcal {P}\cap \mathbb {Z}^d|\) is a polynomial in the integer variable t. Chapoton (2016) proved that, given a fixed integral form \(\lambda : \mathbb {Z}^d \rightarrow \mathbb {Z}\) , there exists a polynomial \(\operatorname {cha}_\mathcal {P}^\lambda (q,x) \in \mathbb {Q}(q)[x]\) such that the refined enumeration function \(\sum _{ \textbf{m}\in t \mathcal {P}} q^{ \lambda (\textbf{m}) }\) equals the evaluation \(\operatorname {cha}_\mathcal {P}^\lambda (q, [t]_q)\) where, as usual, \([t]_q:= \frac{ q^t - 1 }{ q-1 }\) ; naturally, for \(q=1\) we recover the Ehrhart polynomial. Our motivating goal is to view Chapoton’s work through the lens of Brion’s Theorem (1988), which expresses the integer-point structure of a given polytope via that of its vertex cones. It turns out that this viewpoint naturally yields various refinements and extensions of Chapoton’s results, including explicit formulas for \(\operatorname {cha}_\mathcal {P}^\lambda (q,x)\) , its leading coefficient, and its behavior as \(t \rightarrow \infty \) . We also prove an analogue of Chapoton’s structural and reciprocity theorems for rational polytopes (i.e., with vertices in \(\mathbb {Q}^d\) ).
