Israel Journal of Mathematics

, Volume 118, Issue 1, pp 61–82 | Cite as

Center conditions II: Parametric and model center problems

  • M. Briskin
  • J. -P. Francoise
  • Y. Yomdin


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.


Model Problem Center Condition Common Zero Composition Condition Moment Sequence 
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Copyright information

© Hebrew University 2000

Authors and Affiliations

  1. 1.Jerusalem College of EngineeringJerusalemIsrael
  2. 2.Département de MathématiquesUniversité de Paris VI, U.F.R. 920, 46-56ParisFrance
  3. 3.Department of Theoretical MathematicsThe Weizmann Institute of ScienceRehovotIsrael

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