Efficient Rational Creative Telescoping

TL;DR

This paper introduces an efficient algorithm for computing minimal telescopers for rational functions of two variables, improving performance and handling complexity better than previous reduction-based methods.

Contribution

The paper presents a novel algorithm that computes minimal telescopers independently of certificates and uses a compact representation to manage complexity effectively.

Findings

Algorithm has polynomial-time complexity in certificate expansion

Implementation in Maple demonstrates superior practical performance

Handles expression swell efficiently without full expansion

Abstract

We present a new algorithm to compute minimal telescopers for rational functions in two discrete variables. As with recent reduction-based approaches, our algorithm has the important feature that the computation of a telescoper is independent of its certificate. In addition, our algorithm uses a compact representation of the certificate, which allows it to be easily manipulated and analyzed without knowing the precise expanded form. This representation hides potential expression swell until the final (and optional) expansion, which can be accomplished in time polynomial in the size of the expanded certificate. A complexity analysis, along with a Maple implementation, indicates that our algorithm has better theoretical and practical performance than the reduction-based approach in the rational case.