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：设多面体$P$由系统$A x \leq b$定义。我们考虑计算$P$内部整点数量的问题，假设$P$是$\Delta$-模的，即若$A$的所有秩子行列式的绝对值被$\Delta$界定，则称多面体$P$为$\Delta$-模的。我们提出一种新的FPT算法，其参数化由$\Delta$和$P$中最大顶点数决定，其中最大值取遍所有右端向量$b$。我们证明，对于$\Delta$-模问题，该算法比A. Barvinok等人的方法更高效。为此，我们不直接计算通常用于该问题的$P \cap Z^n$的短有理生成函数，而是采用动态规划原理计算其特定表示形式——即依赖于单个变量的指数级数。我们完全未借助Barvinok的单模符号分解技术。利用新的复杂度上界，我们研究了若干可能具有独立意义的特例。例如，我们给出了计算$\Delta$-模单形及类似具有$n + O(1)$个面的多面体中整点数量的FPT算法。作为特例，对于任意固定的$m$，我们给出一个FPT算法，用于计数无界$m$维$\Delta$-模子集和问题的解。