# Algorithms to Compute Discrete Residues of a Rational Function

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## Abstract

The classical notion of residue, for a rational function with complex coefficients, is a powerful and ubiquitous tool, having applications in many different areas. For example: Complex Analysis, Physics, Number Theory, Differential Equations, and Combinatorics, to name a few. In the last decade several new notions of discrete residues have been developed by different researchers, all of which have in common the following obstruction-theoretic feature: a given rational function f (x) is “special” (e.g., rationally integrable, or rationally summable, or rationally q-summable) if and only if all of its corresponding residues are zero. All of these notions of residue (both the classical one and also its discrete variants) are originally defined in terms of a complete partial fraction decomposition of the given rational function f (x), which is too expensive to carry out in practice due to the high computational cost of finding the complete factorization of the denominator. The main contribution of this dissertation is the development of an efficient factorization-free algorithm to compute the discrete residues of a rational function.