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. It gives the fastest way to calculate $E_P(y)$ in certain scenarios. The most important cases are the following: 1) We show that, for the parametric polyhedron $P_y$ defined by a standard-form system $A x = y,\, x \geq 0$ with a fixed number of equalities, the function $E_P(y)$ can be represented by a polynomial-time computable function. In turn, such a representation of $E_P(y)$ can be constructed by an $poly\bigl(n, \|A\|_{\infty}\bigr)$-time algorithm; 2) Assuming again that the number of equalities 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 $E_P$ is more efficient than other known approaches, if $A$ has bounded elements, especially if it 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 $E_P(y) = |P_{y} \cap Z^{n_x}|$。我们提出 $E_P(y)$ 的一种新表示形式，称为**具有周期系数的分段阶梯多项式**，该形式推广了分段阶梯多项式及整数/有理 Ehrhart 拟多项式。在某些场景下，这为计算 $E_P(y)$ 提供了最快的方法。最重要的案例包括：1）对于由标准型系统 $A x = y,\, x \geq 0$ 定义的参数多面体 $P_y$，当等式数量固定时，我们证明 $E_P(y)$ 可由多项式时间可计算函数表示；进而，此类 $E_P(y)$ 的表示可通过 $poly\bigl(n, \|A\|_{\infty}\bigr)$ 时间算法构造；2）在等式数量固定的假设下，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 $A$ 的元素有界时，我们的 $E_P$ 表示比其他已知方法更高效，若 $A$ 还具有稀疏性则优势更显著。此外，我们探讨了该方法在编译器优化领域的潜在应用。在程序代码满足某些“自然”假设的条件下，我们的方法具有最快的复杂度界限。