Differential Galois Theory and non-Integrability of Planar Polynomial Vector Fields

TL;DR

This paper uses differential Galois theory and variational equations to establish necessary conditions for the integrability of planar polynomial vector fields, linking non-integrability to higher transcendental functions.

Contribution

It introduces a systematic method to determine non-integrability of polynomial vector fields using the Risch algorithm and variational equations.

Findings

Necessary conditions for integrability are derived from the elementary nature of primitives.

The method connects non-integrability with higher transcendental functions like the error function.

Illustrative examples demonstrate the application of the theoretical results.

Abstract

We study a necessary condition for the integrability of the polynomials fields in the plane by means of the differential Galois theory. More concretely, by means of the variational equations around a particular solution it is obtained a necessary condition for the existence of a rational first integral. The method is systematic starting with the first order variational equation. We illustrate this result with several families of examples. A key point is to check wether a suitable primitive is elementary or not. Using a theorem by Liouville, the problem is equivalent to the existence of a rational solution of a certain first order linear equation, the Risch equation. This is a classical problem studied by Risch in 1969, and the solution is given by the "Risch algorithm". In this way we point out the connection of the non integrablity with some higher transcendent functions, like the error function.