The Shift Equivalence Testing (SET) of polynomials is deciding whether two polynomials $p(x_1, \ldots, x_m)$ and $q(x_1, \ldots, x_m)$ satisfy the relation $p(x_1 + a_1, \ldots, x_m + a_m) = q(x_1, \ldots, x_m)$ for some $a_1, \ldots, a_m$ in the coefficient field. The SET problem is one of basic computational problems in computer algebra and algebraic complexity theory, which was reduced by Dvir, Oliveira and Shpilka in 2014 to the Polynomial Identity Testing (PIT) problem. This paper presents a general scheme for designing algorithms to solve the SET problem which includes Dvir-Oliveira-Shpilka's algorithm as a special case. With the algorithms for the SET problem over integers, we 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 isotropy groups of polynomials introduced by Sato in 1960s. Our results can be used to detect the applicability of the Wilf-Zeilberger method to multivariate rational functions.

翻译：多项式的移位等价性检验（SET）旨在判定两个多项式 $p(x_1, \ldots, x_m)$ 和 $q(x_1, \ldots, x_m)$ 是否满足关系 $p(x_1 + a_1, \ldots, x_m + a_m) = q(x_1, \ldots, x_m)$，其中 $a_1, \ldots, a_m$ 属于系数域。SET问题是计算机代数与代数复杂性理论中的基本计算问题之一，由Dvir、Oliveira和Shpilka于2014年将其归约为多项式恒等检验（PIT）问题。本文提出一种通用的算法设计框架用于求解SET问题，该框架包含Dvir-Oliveira-Shpilka算法作为特例。借助整数域上的SET问题算法，我们完整解决了多元有理函数符号求和中的两个挑战性问题：有理可和性问题与多元有理函数伸缩算子的存在性问题。我们的方法基于Sato于1960年代引入的多项式迷向群结构。研究结果可用于判定Wilf-Zeilberger方法在多元有理函数中的适用性。