Symbolic Computations of First Integrals for Polynomial Vector Fields

TL;DR

This paper introduces new algorithms for computing various types of first integrals of polynomial vector fields, extending previous methods to more complex cases and demonstrating improved efficiency through implementation and examples.

Contribution

It generalizes the extactic curve approach to Darbouxian, Liouvillian, and Riccati cases, providing new algorithms with better complexity for finding first integrals.

Findings

Probabilistic algorithm complexity is $ ilde{O}(N^{ ext{w}+1})$

Algorithms successfully implemented in Maple

Examples demonstrate efficiency and practicality

Abstract

In this article we show how to generalize to the Darbouxian, Liouvillian and Riccati case the extactic curve introduced by J. Pereira. With this approach, we get new algorithms for computing, if it exists, a rational, Darbouxian, Liouvillian or Riccati first integral with bounded degree of a polynomial planar vector field. We give probabilistic and deterministic algorithms. The arithmetic complexity of our probabilistic algorithm is in $\tilde{\mathcal{O}}(N^{\omega+1})$, where $N$ is the bound on the degree of a representation of the first integral and $\omega \in [2;3]$ is the exponent of linear algebra. This result improves previous algorithms. Our algorithms have been implemented in Maple and are available on authors' websites. In the last section, we give some examples showing the efficiency of these algorithms.