Briskin, M., Francoise, J.-. & Yomdin, Y. Isr. J. Math. (2000) 118: 83. doi:10.1007/BF02803517

Abstract

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 (*).

Folowing [7], we consider a parametric version of this condition: an equation (**)y’=p(x)y^{2} +εq(x)y^{3}p, q as above, ε ∈ ℂ, is said to have a parametric center, if for any ɛ and for any solutiony(ɛ,x) of (**)y(ɛ, 0)≡y(ɛ, 1)..

We give another proof of the fact, shown in [6], 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 generalize a “canonical representation” ofm_{k}(x) given in [7]. On this base we prove in some additional cases a composition conjecture, stated in [6, 7] for a parametric center problem.