A computational approach to rational summability and its applications via discrete residues
Carlos E. Arreche, Hari P. Sitaula
TL;DR
This work develops a practical residue-theoretic framework for rational summability of rational functions, introducing a gcd- and linear-algebra-centric method to compute discrete residues without full partial fraction factorization. By iteratively applying Hermite reduction to reduce to simple poles, and then performing a shift-free, squarefree reduction, the authors obtain a K-rational representation of discrete residues as pairs $(B_k,D_k)$ that encode all obstruction data. They extend the method to serial settings and telescoping, defining spaces $V(\mathbf{f})$ and $W(\mathbf{f})$ to capture linear and differential-relational structures among multiple inputs, with clear connections to Galois theories of difference equations. The paper also reports preliminary Maple implementations and timings, and provides several explicit examples to illustrate how the discrete-residue data dictates summability and telescoping behavior. Overall, the approach yields efficient obstruction checks and enables new algorithmic avenues for studying difference equations, their recurrences, and associated Galois groups in a computation-friendly way.
Abstract
A rational function $f(x)$ is rationally summable if there exists a rational function $g(x)$ such that $f(x)=g(x+1)-g(x)$. Detecting whether a given rational function is summable is an important and basic computational subproblem that arises in algorithms to study diverse aspects of shift difference equations. The discrete residues introduced by Chen and Singer in 2012 enjoy the obstruction-theoretic property that a rational function is summable if and only if all its discrete residues vanish. However, these discrete residues are defined in terms of the data in the complete partial fraction decomposition of the given rational function, which cannot be accessed computationally in general. We explain how to efficiently compute (a rational representation of) the discrete residues of any rational function, relying only on gcd computations, linear algebra, and a black box algorithm to compute the autodispersion set of the denominator polynomial. We also explain how to apply our algorithms to serial summability and creative telescoping problems, and how to apply these computations to compute Galois groups of difference equations.