Let a polytope $P$ be defined by a system $A x \leq b$. We consider the problem of counting the number of integer points inside $P$, assuming that $P$ is $\Delta$-modular, where the polytope $P$ is called $\Delta$-modular if all the rank sub-determinants of $A$ are bounded by $\Delta$ in the absolute value. We present a new FPT-algorithm, parameterized by $\Delta$ and by the maximal number of vertices in $P$, where the maximum is taken by all r.h.s. vectors $b$. We show that our algorithm is more efficient for $\Delta$-modular problems than the approach of A. Barvinok et al. To this end, we do not directly compute the short rational generating function for $P \cap Z^n$, which is commonly used for the considered problem. Instead, we use the dynamic programming principle to compute its particular representation in the form of exponential series that depends on a single variable. We completely do not rely to the Barvinok's unimodular sign decomposition technique. Using our new complexity bound, we consider different special cases that may be of independent interest. For example, we give FPT-algorithms for counting the integer points number in $\Delta$-modular simplices and similar polytopes that have $n + O(1)$ facets. As a special case, for any fixed $m$, we give an FPT-algorithm to count solutions of the unbounded $m$-dimensional $\Delta$-modular subset-sum problem.

翻译：设多胞体$P$由系统$A x \leq b$定义。我们考虑计算$P$内部整点数量的计数问题，假设$P$是Δ-模的，其中多胞体$P$被称为Δ-模的当且仅当矩阵$A$的所有秩子行列式的绝对值均不超过Δ。我们提出一种新的FPT算法，其参数由Δ和$P$的最大顶点数决定（其中最大值取遍所有右端向量$b$）。我们证明，对于Δ-模问题，该算法比A. Barvinok等人的方法更为高效。为此，我们并未直接计算通常用于该问题的$P \cap Z^n$的短有理生成函数，而是利用动态规划原理，计算其以依赖于单个变量的指数级数形式表示的特殊表示式。我们完全不依赖Barvinok的酉模符号分解技术。基于新的复杂度界，我们考虑了可能具有独立意义的多种特殊情形。例如，我们给出了计算Δ-模单纯形及具有$n + O(1)$个面的相似多胞体中整点数量的FPT算法。作为特例，对于任意固定的$m$，我们提出了一种FPT算法，用于计算无界$m$维Δ-模子集和问题的解的个数。