Center conditions at infinity for Abel differential equations

Abstract

An Abel differential equation $y’=p(x)y^2 + q(x) y^3$ is said to have a center at a set $A=\{a_1,\dots,a_r\}$ of complex numbers if $y(a_1)=y(a_2)=\dots=y(a_r)$ for any solution $y(x)$ (with the initial value $y(a_1)$ small enough).

The polynomials $p,q$ are said to satisfy the “Polynomial Composition Condition” on $A$ if there exist polynomials $\tilde P$, $\tilde{Q}$ and $W$ such that $P=\int p$ and $Q=\int q$ are representable as $P(x)=\tilde{P}(W(x))$, $Q(x)=\tilde{Q}(W(x))$, and $W(a_1)=W(a_2)=…=W(a_r)$. We show that for wide ranges of degrees of $P$ and $Q$ (restricted only by certain assumptions on the common divisors of these degrees) the composition condition provides a very accurate approximation of the Center one — up to a finite number of configurations not accounted for. To our best knowledge, this is the first “general” (i.e., not restricted to small degrees of $p$ and $q$ or to a very special form of these polynomials) result in the Center problem for Abel equations.

As an important intermediate result we show that “at infinity” (according to an appropriate projectivization of the parameter space) the Center conditions are given by a system of the “Moment equations” of the form $\int^{a_s}_{a_1} P^k q = 0$, $s=2,\dots,r$, $k=0,1,…\; $.