Abstract

A differential equation of the form $y\prime (x)=\; p(x)\; y2+\; q(x)\; y3$ is called an Abel differential equation. The composition conjecture for these kind of differential equations states that these equations have a center if and only if the polynomials p and q satisfy a convenient property.
The Composition Conjecture in a more general context appeared in 1987 and it is due to Alwash and Lloyd. It is known that this conjecture is not true if $p(x)$ and $q(x)$ are polynomial functions in $cos(x)$ and $sin(x)$. The conjecture in its present version was introduced by Briskin, Francoise and Yomdin in 1999. The "only if" part of the Composition Conjecture always holds. Up to now there are no counterexamples to the "if" part of the Composition Conjecture.
Part of this talk is a joint work with Jiazhong Yang.