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 (*).
We introduce a parametric version of this condition: an equation (**)y’=p(x)y2 +εq(x)y3p, 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 momentsmk(1), wheremk(x)=∫0xpk(t)q(t)(dt),P(x)=∫0xp(t)dt. We investigate the structure of zeroes ofmk(x) and on this base prove in some special cases a composition conjecture, stated in , for a parametric center problem.
Model Problem Center Condition Common Zero Composition Condition Moment Sequence
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
This is a preview of subscription content, log in to check access
M. Briskin, J.-P. Francoise and Y. Yomdin,Center conditions, composition of polynomials and moments on algebraic curves, Ergodic Theory and Dynamical Systems19 (1999), 1201–1220.MATHCrossRefMathSciNetGoogle Scholar
M. Briskin, J.-P. Francoise and Y. Yomdin,Center conditions III: More on parametric and model center problems, Israel Journal of Mathematics, this volume.Google Scholar
J. Devlin,Word problem related to periodic solutions of a non-autonomous system, Mathematical Proceedings of the Cambridge Philosophical Society108 (1990), 127–151.MATHMathSciNetCrossRefGoogle Scholar
J. Devlin,Word problems related to derivatives of the displacement map, Mathematical Proceedings of the Cambridge Philosophical Society110 (1991), 569–579.MATHMathSciNetGoogle Scholar
D. V. Widder,The Laplace Transform, Princeton University Press, N.J., 1946.MATHGoogle Scholar
H. S. Wilf and D. Zeilberger,An algorithmic proof theory for hypergeometric (ordinary and “q”) multisum/integral identities, Inventiones Mathematicae108 (1992), 575–633.CrossRefMathSciNetGoogle Scholar