Integrability conditions of a resonant saddle in generalized Liénard-like complex polynomial differential systems

Abstract

We consider a complex differential system with a resonant saddle at the origin. We compute the resonant saddle quantities and using Gröbner bases we find the integrability conditions for such systems up to a certain degree. We also establish a conjecture about the integrability conditions for such systems when they have arbitrary degree.

The first author is partially supported by a MINECO/ FEDER grant number MTM2014-53703-P and an AGAUR (Generalitat de Catalunya) grant number 2014SGR 1204. The second author is partially supported by a MINECO grant MTM2013-40998-P, an AGAUR grant number 2014SGR-568, and the grants FP7-PEOPLE-2012-IRSES 318999 and 316338.