Center conditions II: Parametric and model center problems

We consider an Abel equation (*)y’=p(x)y ² +q(x)y ³ 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 thaty ₀=y(0)≡y(1) for any solutiony(x) of (*). We introduce a parametric version of this condition: an equation (**)y’=p(x)y ² +εq(x)y ³ p, q as above, ℂ, is said to have a parametric center, if for any ε and for any solutiony(ε,x) of (**),y(ε,0)≡y(ε,1). We show that the parametric center condition implies vanishing of all the momentsm _k (1), wherem _k (x)=∫ ₀ ^{x pk} (t)q(t)(dt),P(x)=∫ ₀ ^x p(t)dt. We investigate the structure of zeroes ofm _k (x) and on this base prove in some special cases a composition conjecture, stated in 10], for a parametric center problem. The research of the first and the third author was supported by the Israel Science Foundation, Grant No. 101/95-1 and by the Minerva Foundation.