Dr. Hari Sitaula (Dept. of Mathematical Sciences, MT. Tech. Univ.)

02/20/2025 3:10pm

Abstract:

In 2012 Chen and Singer introduced the notion of discrete residues for rational functions as a complete obstruction to rational sum ability. 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 applications beyond their immediate definition as obstructions to sum ability: to creative telescoping problems, thence to the determination of (differential-)algebraic relations amongst hypergeometric sequences, and subsequently to the computation of (differential) Galois groups of different equations. However, the discrete residues of a rational function are initially defined in terms of its complete partial fraction decomposition. This makes their direct computation, as dictated by their definition, impractical in general, due to the high complexity of completely factoring arbitrary denominator polynomials into linear factors. We develop an algorithm to compute discrete residues of rational functions relying only on gcd computations and linear algebra.