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 ,..., a r } of complex numbers if y(a 1 ) = y(a 2 ) =··· = 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 P, Q and W such that P = ∫ p and Q = ∫ q are representable as P(x) = P(W(x)), Q(x) = 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 ∫ as a1 P k q = 0, s = 2,..., r, k = 0, 1,..

http://annals.math.princeton.edu/wp-content/uploads/annals-v172-n1-p08-p.pdf

Center conditions at infinity for Abel differential equations

Generalizations of the image conjecture and the Mathieu conjecture

Mathieu subspaces of associative algebras

In this paper we give a solution of the following ‘polynomial moment problem’ which arose about

https://www.math.bgu.ac.il/~pakovich/Publications/s.pdf

Solution of the polynomial moment problem

In this paper we give a complete solution of the following "polynomial moment problem" which arose about 10 years ago in connection with Poincare's center-focus problem. For a given polynomial P(z) to describe polynomials Q(z) orthogonal to all powers of P(z) on a segment [a,b].

We consider an Abel equation (*)y’=p(x)y2 +q(x)y3 withp(x), q(x) polynomials inx. A center condition for (*) (closely related to the classical center condition for polynomial vector fields on the plane) is thaty0=y(0)≡y(1) for any solutiony(x) of (*).

M. Briskin

Papers

Center conditions III: Parametric and model center problems