发表机构
Capital Normal University; Beijing Institute of Technology（首都师范大学; 北京理工大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究提出首个固定维度下无需Barvinok分解的多项式时间格点计数算法，基于\texttt{SimpCone[S]}框架和Minkowski定理，可处理Laurent多项式分子，为MacMahon分拆分析提供多项式时间方案。
AI 中文摘要
通过使用常数项操作，我们提出了首个固定维度下不依赖Barvinok unimodular分解的多项式时间格点计数算法。该算法直接作用于由\texttt{SimpCone[S]}框架生成的嵌套根平均形式的有理生成函数。借助基于Minkowski定理的剩余格论证，我们构造了一个短乘子，该乘子可对最外层平均进行精确的非互素拆分。得到的子项被编码为联合根平均，且使用Smith标准型恢复递归结构。两个结构不变量——生成条件和满列独立性——确保递归定义良好，且所有所需的极点交换均有效。对于固定维度的单纯锥，该算法达到递归深度\\(O_d(1+\log\log(2+\ind(\mathcal K^*)))\\)，并生成最多\\((1+\log \ind(\mathcal K^*))^{O_d(1)}\\)个 unimodular 锥生成函数的带符号和。该框架统一处理分子为Laurent多项式的情况，而不仅限于单项式，从而在维度固定时为MacMahon分拆分析提供了多项式时间算法。
英文摘要
By using constant term manipulations, we present the first polynomial-time algorithm for lattice-point counting in fixed dimension that does not rely on Barvinok's unimodular decomposition. Our method starts from partial fraction decompositions and root-of-unity formulas for the lattice-point generating function of a rational simplicial cone. By means of a residue-lattice argument based on Minkowski's theorem, we construct a short multiplier that enables an exact recursive reduction. Smith normal form is then used to normalize the resulting child terms and restore the recursive structure. These transformations are justified by algebraic identities involving roots of binomial equations, while the algorithm itself works with integer exponent data and requires no explicit root computations. More precisely, for a rational simplicial cone \(\mathcal K\) in fixed dimension \(d\), with dual cone \(\mathcal K^*\), the algorithm achieves recursion depth \(O_d(1+\log\log(2+\ind(\mathcal K^*)))\) and produces a signed sum of at most \((1+\log\ind(\mathcal K^*))^{O_d(1)}\) unimodular cone generating functions. The framework uniformly handles numerators that are Laurent polynomials, not merely monomials, thereby giving a polynomial-time algorithm for MacMahon's partition analysis when the dimension is fixed.
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