A recombination algorithm for the decomposition of multivariate rational functions

TL;DR

This paper introduces a deterministic recombination algorithm for decomposing multivariate rational functions using Darboux polynomials, demonstrating improved complexity over previous methods and extending to sparse bivariate cases.

Contribution

The paper presents a novel recombination strategy leveraging Darboux polynomials for rational function decomposition, with complexity analysis and application to sparse bivariate functions.

Findings

The new method improves decomposition complexity compared to previous algorithms.

The approach effectively handles sparse bivariate rational functions.

Complexity bounds are established for the proposed algorithm.

Abstract

In this paper we show how we can compute in a deterministic way the decomposition of a multivariate rational function with a recombination strategy. The key point of our recombination strategy is the used of Darboux polynomials. We study the complexity of this strategy and we show that this method improves the previous ones. In appendix, we explain how the strategy proposed recently by J. Berthomieu and G. Lecerf for the sparse factorization can be used in the decomposition setting. Then we deduce a decomposition algorithm in the sparse bivariate case and we give its complexity