Center and composition conditions for Abel differential equation, and rational curves

Abstract

We consider the Abel Equation $\r'= p(\t)\r^2 + q(\t)\r^3$ (*) with $ p(\t)$, $q(\t)$ - polynomials in $\sin \t$, $\cos \t$. The center problem for this equation (which is closely related to the classical center problem for polynomial vector fields on the plane) is to find conditions on $p$ and $q$ under which all the solutions $\r(\t)$ of this equation are periodic, i.e. $\r(0)=\r(2 \pi)$ for all initial values $\r(0)$. We consider the equation (*) as an equation on the complex plane $\frac{dy}{dz}=p(z)y^2 + q(z) y^3$ (**) with $p$, $q$ -- Laurent polynomials. Then the center condition is that its solution $y(z)$ is a univalued function along the circle $|z|=1$. We study the behavior of the equation (**) under mappings of the complex plane onto Riemann Surfaces. This approach relates the center problem to the algebra of rational functions under composition and to the geometry of rational curves. We obtain the sufficient conditions for the center in the form $\int_{|z|=1} P^i Q^j dP=0$ with $P=\int p$, $Q=\int q$.