Algebraic geometry of Abel differential equation

Abstract

A solution $y(x)$ of an Abel differential equation $(1) \ y^{\prime }=p(x)y^2 + q(x) y^3$ is called “closed” on $[a,b]$ if $y(a)=y(b)$. The equation $(1)$ is said to have a center on $[a,b]$ if all its solutions (with the initial value $y(a)$ small enough) are closed. The problems of counting closed solutions (Smale–Pugh problem) is strongly related to the classical Hilbert 16th problem of bounding the number of limit cycles of a plane polynomial vector field. In turn, the problem of giving conditions on $(p,q,a,b)$ for $(1)$ to have a center on $[a,b]$ is analogous to the classical Poincaré center–focus problem for plane vector fields. It is well known that both in the classical and in the Abel equation cases the center conditions are given by an infinite system of polynomial equations in the parameter space. The complexity of this system presents one of the main difficulties in the center–focus problem, as well as in the bounding of closed trajectories. In recent years two important structures have been related to the center equations for $(1)$: composition algebra and moment vanishing. In the present paper we give an overview of these results (sometimes providing also new ones), stressing their algebraic–geometric interpretation and consequences. The second part of the paper is devoted to a rather detailed study of a specific example of the Abel equation which possesses algebraic solutions. We identify complex closed solutions, stressing their ramification properties. In particular, we analyze the continuation paths along which these solutions become closed.