Efficient Algorithms for Computing Rational First Integrals and Darboux Polynomials of Planar Polynomial Vector Fields

TL;DR

This paper introduces efficient algorithms for computing rational first integrals and Darboux polynomials of planar polynomial vector fields, significantly improving computational complexity over previous methods.

Contribution

It presents novel linear algebra-based algorithms for faster computation of rational first integrals and Darboux polynomials, with both probabilistic and deterministic variants.

Findings

Probabilistic algorithm runs in $igOsoft(N^{2 \omega})$ operations.

Deterministic algorithm runs in $igOsoft(d^2N^{2 \omega+1})$ operations.

Heuristic variant computes rational first integrals in $igOsoft(N^{\omega+2})$ operations.

Abstract

We present fast algorithms for computing rational first integrals with bounded degree of a planar polynomial vector field. Our approach is inspired by an idea of Ferragut and Giacomini. We improve upon their work by proving that rational first integrals can be computed via systems of linear equations instead of systems of quadratic equations. This leads to a probabilistic algorithm with arithmetic complexity $\bigOsoft(N^{2 \omega})$ and to a deterministic algorithm solving the problem in $\bigOsoft(d^2N^{2 \omega+1})$ arithmetic operations, where $N$ denotes the given bound for the degree of the rational first integral, and where $d \leq N$ is the degree of the vector field, and $\omega$ the exponent of linear algebra. We also provide a fast heuristic variant which computes a rational first integral, or fails, in $\bigOsoft(N^{\omega+2})$ arithmetic operations. By comparison, the best previous algorithm uses at least $d^{\omega+1}\, N^{4\omega +4}$ arithmetic operations. We then show how to apply a similar method to the computation of Darboux polynomials. The algorithms are implemented in a Maple package which is available to interested readers with examples showing its efficiency.