Reduction of symbolic first integrals of planar vector fields

TL;DR

This paper introduces algorithms to simplify symbolic first integrals of planar polynomial vector fields, enabling the determination of minimal class integrals and rational integrability, with specific procedures for complex cases involving elliptic factors.

Contribution

The paper presents novel algorithms for reducing the complexity class of symbolic first integrals and for testing rational integrability in planar vector fields.

Findings

Algorithms successfully reduce first integrals to lower classes.

Method can identify non-rational integrals in most cases.

Special procedures handle complex Darbouxian cases with elliptic factors.

Abstract

Consider a planar polynomial vector field $X$, and assume it admits a symbolic first integral $\mathcal{F}$, i.e. of the $4$ classes, in growing complexity: Rational, Darbouxian, Liouvillian and Riccati. If $\mathcal{F}$ is not rational, it is sometimes possible to reduce it to a simpler class first integral. We will present algorithms to reduce symbolic first integral to a lower complexity class. These algorithms allow to find the minimal class first integral and in particular to test the existence of a rational first integral except in the case where $\mathcal{F}$ is a $k$-Darbouxian first integral without singularities and $k\in\{2,3,4,6\}$. In this case, several examples are built and a procedure is presented which however requires the computation of elliptic factors in the Jacobian of a superelliptic curve.