In this paper, we consider the counting function $E_P(y) = |P_{y} \cap Z^{n_x}|$ for a parametric polyhedron $P_{y} = \{x \in R^{n_x} \colon A x \leq b + B y\}$, where $y \in R^{n_y}$. We give a new representation of $E_P(y)$, called a \emph{piece-wise step-polynomial with periodic coefficients}, which is a generalization of piece-wise step-polynomials and integer/rational Ehrhart's quasi-polynomials. In terms of the computational complexity, our result gives the fastest way to calculate $E_P(y)$ in certain scenarios. The most remarkable cases are the following: 1) Consider a parametric polyhedron $P_y$ defined by a standard-form system $A x = y,\, x \geq 0$ with a fixed number of equalities. We show that there exists an $poly\bigl(n, \|A\|_{\infty}\bigr)$ preprocessing-algorithm that returns a polynomial-time computable representation of $E_P(y)$. That is, $E_(y)$ can be computed by a polynomial-time algorithm for any given $y \in Q^k$; 2) Again, assuming that the co-dimension is fixed, we show that integer/rational Ehrhart's quasi-polynomials of a polytope can be computed by FPT-algorithms, parameterized by sub-determinants of $A$ or its elements; 3) Our representation of $E_P(y)$ is more efficient than other known approaches, if the matrix $A$ has bounded elements, especially if the matrix $A$ is sparse in addition; Additionally, we provide a discussion about possible applications in the area of compiler optimization. In some "natural" assumptions on a program code, our approach has the fastest complexity bounds.

翻译：本文研究参数多面体 $P_{y} = \{x \in R^{n_x} \colon A x \leq b + B y\}$（其中 $y \in R^{n_y}$）的计数函数 $E_P(y) = |P_{y} \cap Z^{n_x}|$。我们给出了 $E_P(y)$ 的一种新表示，称为“具有周期系数的分段阶梯多项式”，这是对分段阶梯多项式和整数/有理 Ehrhart 拟多项式的推广。在计算复杂度方面，我们的结果在特定场景下提供了计算 $E_P(y)$ 的最快方法。最显著的案例包括：1) 考虑由标准型系统 $A x = y,\, x \geq 0$ 定义的参数多面体 $P_y$，其中等式数量固定。我们证明存在一个 $poly\bigl(n, \|A\|_{\infty}\bigr)$ 预处理算法，可返回 $E_P(y)$ 的多项式时间可计算表示。即，对于任意给定的 $y \in Q^k$，$E_(y)$ 可由多项式时间算法计算；2) 同样在余维数固定的假设下，我们证明多面体的整数/有理 Ehrhart 拟多项式可通过以 $A$ 的子行列式或其元素为参数的FPT算法计算；3) 当矩阵 $A$ 元素有界（尤其当矩阵 $A$ 稀疏）时，我们的 $E_P(y)$ 表示比其他已知方法更高效。此外，我们讨论了该成果在编译器优化领域的潜在应用。针对程序代码的某些“自然”假设，我们的方法具有最快的复杂度界。