发表机构
Wolfram Research（Wolfram Research）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文为多项式基调和数构建了一个保守的有限约化库，涵盖多种约化机制，并提供Mathematica包，含约670条规则，支持仿射与普通调和数的高效计算。
AI 中文摘要
我们为多重多项式基调和数（即每个求和层分母字母为一元多项式的严格着色嵌套和）开发了一个有限约化库。仿射和普通有限多重调和数作为低复杂度的子类出现，分别对应于一次多项式字母和普通字母$k$。主要机制包括局部归一化、有理单层下降、欧几里得除法、部分分式、将多项式字母分解为仿射字母、二次分裂、空层和多项式分子层的精确求和、仿射平移和格约化、阶梯和补变换、重复层牛顿约化、弱到严格对角分解以及终端普通调和数约化。随附的Mathematica包提供了一个紧凑的可执行约化库，用于多项式基、仿射和普通有限调和数对象，并在补充数据挖掘笔记本中记录了多个已检查的示例。当前补充规则清单索引了约670条约化、保护和归一化条目，其中约160条为命名族级条目。该库有意保持保守：规则仅在明确假设下应用，例如有限求和范围内无极点、部分分式下降的整数幂假设、分支安全缩放、在允许的系数扩展上的有限分解，以及对于伸缩求和，可验证的证书。
英文摘要
We develop a finite reduction library for multiple polynomial-base harmonic numbers, strict colored nested sums in which the denominator letters at each summation level are univariate polynomials. The affine and ordinary finite multiple harmonic numbers appear as lower-complexity subclasses, corresponding respectively to degree-one polynomial letters and to the ordinary letter $k$. The main mechanisms include local normalization, rational single-level descent, Euclidean division, partial fractions, factorization of polynomial letters into affine letters, quadratic splitting, exact summation of empty and polynomial-numerator levels, affine shift and lattice reductions, staircase and complement transformations, repeated-level Newton reductions, weak-to-strict diagonal decompositions, and terminal ordinary harmonic-number reductions. The accompanying Mathematica package provides a compact executable reduction library for polynomial-base, affine, and ordinary finite harmonic-number objects, with many checked examples recorded in a supplementary data-mine notebook. The current supplementary rule inventory indexes roughly 670 reduction, guard, and normalization entries, of which about 160 are named family-level entries. The library is intentionally conservative: rules are applied only under explicit hypotheses, such as absence of poles on the finite summation range, integer-power assumptions for partial-fraction descent, branch-safe scaling, finite factorization over an allowed coefficient extension, and, for telescoping, a verifiable certificate.
Comments16 pages, no figures
