Ehrhart Quasi-Polynomials of Rational Simple Polytopes

• Ehrhart quasi-polynomials are functions that count lattice points in rationally dilated simple polytopes, exhibiting periodic and piecewise-polynomial behavior.
• They detail the combinatorial structure of rational polytopes through methods like vertex cone decompositions, Barnes polynomials, and inclusion-exclusion formulas.
• Efficient computation via algorithms such as Barvinok’s method connects these quasi-polynomials to applications in toric geometry and discrete optimization.

A rational simple polytope is a convex polytope in $\mathbb{R}^d$ with rational vertices such that exactly $d$ facets meet at each vertex. The lattice-point counting function

$L_P(r) = \#(r P \cap \mathbb{Z}^d)$

for $r \in \mathbb{Q}_{\ge 0}$ generalizes the classical case of integer dilates and encodes the discrete geometric structure of $P$ under rational scaling. The resulting function is described by the theory of Ehrhart quasi-polynomials, which in the rational setting exhibits rich periodic, piecewise-polynomial, and combinatorial behavior strongly influenced by the denominators of $P$ and the dilations.

1. Rational Simple Polytopes and Rational Ehrhart Quasi-Polynomials

Let $P \subset \mathbb{R}^d$ be a simple rational $d$-polytope, i.e., all vertices of $P$ are in $\mathbb{Q}^d$ and at each vertex exactly $d$ facets meet. The minimal positive rational $q(P)$ such that $q(P) P$ is integral is called the rational denominator of $P$ (Linke, 2010).

For every $r \in \mathbb{Q}_{\ge 0}$, the rational dilation $rP := \{ r x : x \in P \}$ may not be a lattice polytope, and the behavior of $L_P(r)$ reflects the interaction of the scaling factor with the underlying lattice. The lattice-point counting function $L_P(r)$ is then a rational quasi-polynomial: $L_P(r) = \sum_{i=0}^d c_i(r) r^i,$ where each $c_i(r)$ is a periodic, piecewise-polynomial function of $r$ with period equal to the $i$-index $_i(P)$, defined as the minimal positive $r$ such that every $i$-face $F \subset P$ satisfies $r\,\mathrm{aff}(F)\cap\mathbb{Z}^n \neq\emptyset$ (Linke, 2010).

2. Structure and Properties of Rational Quasi-Polynomials

For a rational simple polytope $P$ of dimension $d$ and rational denominator $q(P)$, $L_P(r)$ is a degree-$d$ rational quasi-polynomial of period $q(P)$. Each coefficient $c_i(r)$ is itself piecewise-polynomial of degree $d-i$, with periodicity given by the $i$-index $_i(P)$ (Linke, 2010). The function is locally polynomial away from finitely many discontinuities, occurring precisely at values of $r$ where the dilated facets cross lattice hyperplanes.

In symbols, on each interval of constancy, $c_i(r)$ has a representation

$$c_i(r) = \sum_{j=0}^{d-i} c_{i,j}^{(m)} r^{j}\qquad \text{for } r\in I_m\subset [0,q(P)),$$

with the intervals $I_m$ arising from the crossing patterns of the facets with the lattice.

The leading coefficient $c_d(r)$ is constant, equal to $\operatorname{Vol}(P)$, while lower coefficients exhibit explicit piecewise-polynomial structure reflecting the combinatorics of the faces of $P$ (Linke, 2010, Beck et al., 2021).

3. Closed-Form Expressions in the Simple Case

Simple polytopes admit closed expressions for the rational quasi-polynomial via explicit inclusion-exclusion or local primary decomposition. For the rational simplex $\Delta = \operatorname{conv}(0, v_1, \ldots, v_d)$ with $v_j\in\mathbb{Q}^d$, the Ehrhart quasi-polynomial for rational $r$ is

$$L_{\Delta}(r) = \sum_{i=0}^d \left(\sum_{1\leq j_1 < \cdots < j_i \leq d} \det(v_{j_1}, \ldots, v_{j_i}) \right) \frac{r^i}{i!} + (\text{lower-order periodic terms}),$$

with the combinatorial minors possibly themselves periodic (Linke, 2010).

For the rational cube $P = [0, a_1/b_1] \times \cdots \times [0, a_n/b_n]$, it follows that

$$L_P(r) = \prod_{j=1}^n \bigl( \lfloor r a_j/b_j \rfloor + 1 \bigr),$$

and each coefficient $c_i(r)$ is an explicit sum of polynomials in fractional parts $\{ r B_I \}$ for subsets $I$ of the indices (Linke, 2010).

For general simple rational polytopes, Brion's theorem provides a decomposition of $L_P(r)$ as a sum over the vertex cones, each being a rational generating function whose periodic and piecewise behavior can be accessed via finite Fourier analysis or the calculation of lattice-point transforms (Beck et al., 28 May 2025, Robins, 18 Jan 2026). In particular, Barnes polynomials and discrete moments of half-open parallelepipeds at each vertex allow a direct computation: $$L_P(t) = \frac{(-1)^d}{d!} \sum_{v\in V(P)} \sum_{k=0}^d \binom{d}{k} B_k(t \langle v, z \rangle; a_v) \sum_{q \in \mathbb{Z}^d - t v \bmod \Pi_v} \langle q, z \rangle^{d-k},$$ where $B_k$ denotes Barnes polynomials and $\Pi_v$ the fundamental parallelepiped at vertex $v$ (Robins, 18 Jan 2026).

4. Differentiation and Functional Relations Among Coefficients

A key structural property is that the coefficients of the rational quasi-polynomial satisfy derivation relations. For all $i=0,\ldots,d-1$,

$\frac{d}{dr} c_i(r) = - (i+1) c_{i+1}(r)$

away from discontinuities, matching the formal derivative of $L_P(r)$. This forms a triangular system tightly constraining the possible behavior of the coefficients (Linke, 2010, Robins, 18 Jan 2026).

This ODE structure extends to moments of higher-degree weights and provides a powerful handle for analytic and algorithmic manipulation (Robins, 18 Jan 2026).

5. Computation, Algorithms, and Complexity

For fixed dimension, explicit computation of the quasi-polynomial is feasible via Barvinok's algorithm for integer Ehrhart quasi-polynomials of rational polytopes, or by finite-difference recurrence. The transition to rational dilation $r = a/b$ leverages computation of the integer-dilate Ehrhart quasi-polynomial for $P/b$, with

$c_i(a/b) = G_i(P/b, a) b^i,$

where $G_i(P/b, \cdot)$ is the $i$th coefficient in the integral quasi-polynomial expansion (Linke, 2010). The periods for $c_i$ are controlled by the $i$-indices of $P$ and their scaling (Linke, 2010).

Efficient computation of step-polynomial quasi-coefficients is possible for the highest-degree terms even as $d$ grows, using patched generating-function techniques based on local cone decompositions and short rational expressions for half-open parallelepipeds (Baldoni et al., 2010). The essential complexity reduces to summing polynomially many terms over lattice fragments determined by the denominators and face structure—conjecturally polynomial time for fixed $d$ and fixed degree in $r$ (Robins, 18 Jan 2026).

6. Period Collapse and Piecewise-Polynomial Phenomena

While the generic period of the rational Ehrhart quasi-polynomial is the least common multiple of the denominators of the vertices (the denominator of the polytope), period collapse can occur, particularly for families with special symmetry or combinatorial structure (McAllister et al., 2015, Fernandes et al., 2021). In the plane, every pair $(r,s)$ of positive integers arises as the minimal periods of the constant and linear coefficients $(s_0,s_1)$, with the quadratic (leading) term always constant (McAllister et al., 2015). For higher-dimensional specific classes (e.g., polytopes attached to graphs), coset decompositions can force smaller-than-expected period to arise (Fernandes et al., 2021).

The piecewise-polynomial nature of the coefficients is tied to the interaction between dilation and the lattice: at specific $r$ (rational thresholds where facets cross lattice points), the polynomial structure in $c_i(r)$ may jump (Linke, 2010). On fixed intervals of $r$ between such thresholds, the coefficients are polynomial.

7. Extensions and Applications

The rational Ehrhart quasi-polynomial framework applies to parametric families: if a simple rational polytope has vertices that vary as rational functions of a parameter, the lattice point count remains an eventual quasi-polynomial (Chen et al., 2010). Applications include the study of (weighted) enumeration of discrete structures parameterized by rational polytopes, connections to toric geometry, and classification problems in combinatorics.

Rational simple polytopes occur naturally in zonotopal tilings, Coxeter arrangements, and combinatorics of alcoved polytopes, where the explicit knowledge of the Ehrhart quasi-polynomial and its constituent structure enables enumeration in algebraic, geometric, and topological contexts (Bullock et al., 2024, Ardila et al., 2020).

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References:

• (Linke, 2010) Rational Ehrhart quasi-polynomials
• (Beck et al., 2021) Rational Ehrhart Theory
• (Baldoni et al., 2010) Computation of the highest coefficients of weighted Ehrhart quasi-polynomials
• (Robins, 18 Jan 2026) Ehrhart quasi-polynomials via Barnes polynomials and discrete moments of parallelepipeds
• (McAllister et al., 2015) Ehrhart quasi-period collapse in rational polygons
• (Fernandes et al., 2021) On the period collapse of a family of Ehrhart quasi-polynomials
• (Bullock et al., 2024) The Ehrhart series of alcoved polytopes
• (Chen et al., 2010) Generalized Ehrhart polynomials
• (Beck et al., 28 May 2025) A Closer Look at Chapoton's q-Ehrhart Polynomials
• (Ardila et al., 2020) The Arithmetic of Coxeter Permutahedra