  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.
