In 2012 Chen and Singer introduced the notion of discrete residues for rational functions as a complete obstruction to rational summability. More explicitly, for a given rational function f(x), there exists a rational function g(x) such that f(x) = g(x+1) - g(x) if and only if every discrete residue of f(x) is zero. Discrete residues have many important further applications beyond summability: to creative telescoping problems, thence to the determination of (differential-)algebraic relations among hypergeometric sequences, and subsequently to the computation of (differential) Galois groups of difference equations. However, the discrete residues of a rational function are defined in terms of its complete partial fraction decomposition, which makes their direct computation impractical due to the high complexity of completely factoring arbitrary denominator polynomials into linear factors. We develop a factorization-free algorithm to compute discrete residues of rational functions, relying only on gcd computations and linear algebra.

翻译：2012年，Chen与Singer引入了有理函数离散留数的概念，将其作为有理可和性的完全障碍。具体而言，对于给定有理函数f(x)，存在有理函数g(x)使得f(x)=g(x+1)-g(x)当且仅当f(x)的每个离散留数均为零。除可和性外，离散留数还具有许多重要应用：可应用于创造性的伸缩问题，进而判定超几何序列间的（微分）代数关系，并最终用于计算差分方程的（微分）伽罗瓦群。然而，有理函数的离散留数定义依赖于其完整部分分式分解，由于完全分解任意分母多项式为线性因子的复杂度极高，导致直接计算不切实际。本文提出一种无需因式分解的算法来计算有理函数离散留数，该算法仅依赖最大公因式计算与线性代数。