多元有理函数的符号求和
Symbolic Summation of Multivariate Rational Functions
AI总结：
本文针对多元有理函数符号求和，基于 Sato 迷向群将问题归约为多项式移位等价性检验，完整解决了有理可和性与望远镜存在性问题。
摘要：
符号求和作为符号计算的一个活跃研究课题，为求值和化简来自数学、计算机科学、物理学及其他领域的各类求和提供了高效的算法工具。现有的大多数符号求和算法主要适用于单变量输入问题。符号计算中的一项长期工程是开发用于多元函数符号求和的理论、算法和软件。本文将完整解决多元有理函数符号求和中的两个具有挑战性的问题，即有理可和性问题以及多元有理函数望远镜（telescopers）的存在性问题。我们的方法基于多项式的 Sato 迷向群结构，这使我们能够将这些问题归约为检验多项式的移位等价性。我们的结果为微分形式上 Picard 问题的离散类似物提供了完整解答，并可用于检测 Wilf-Zeilberger 方法对多元有理函数的适用性。
英文摘要：
Symbolic summation as an active research topic of symbolic computation provides efficient algorithmic tools for evaluating and simplifying different types of sums arising from mathematics, computer science, physics and other areas. Most of existing algorithms in symbolic summation are mainly applicable to the problem with univariate inputs. A long-term project in symbolic computation is to develop theories, algorithms and software for the symbolic summation of multivariate functions. This paper will give complete solutions to two challenging problems in symbolic summation of multivariate rational functions, namely the rational summability problem and the existence problem of telescopers for multivariate rational functions. Our approach is based on the structure of Sato's isotropy groups of polynomials, which enables us to reduce the problems to testing the shift equivalence of polynomials. Our results provide a complete solution to the discrete analogue of Picard's problem on differential forms and can be used to detect the applicability of the Wilf-Zeilberger method to multivariate rational functions.
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