Polynomial and rational first integrals for planar homogeneous polynomial differential systems

Abstract

In this paper we find necessary and suficient conditions in order that a planar homogeneous polynomial differential system has a polynomial or rational first integral. We apply these conditions to linear and quadratic homogeneous polynomial differential systems.

The first and second authors are partially supported by a MICINN/ FEDER grant number MTM2011-22877 and by a AGAUR (General- itat de Catalunya) grant number 2009SGR 381. The third author is partially supported by a MICINN/ FEDER grant number MTM2008- 03437, by a AGAUR grant number 2009SGR 410 and by ICREA Academia