Israel Journal of Mathematics

, Volume 118, Issue 1, pp 83–108

Center conditions III: Parametric and model center problems


    • Jerusalem College of Engineering
  • J. -P. Francoise
    • Département de MathématiquesUniversité de Paris VI, U.F.R. 920, 46-56
  • Y. Yomdin
    • Department of Theoretical MathematicsThe Weizmann Institute of Science

DOI: 10.1007/BF02803517

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


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.

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© Hebrew University 2000