Center conditions at infinity for Abel differential equations. (English) Zbl 1216.34025

It is said that the Abel differential equation
\[ y'=p(x) y^2+q(x) y^3,\tag{1} \]
where \(p\) and \(q\) are polynomials, has a center at a set \(A=\{a_1,\dots, a_r\}\) of complex numbers if \(y(a_1)=\dots=y(a_r)\) for any solution \(y(x)\) with the initial value \(y(a_1)\) sufficiently small. The center problem for (1) is to give necessary and sufficient conditions for the Abel equation (1) to have a center in terms of \(p\) and \(q\). Although the original interest in the center problem for the Abel equation comes from the study of the center problem for polynomial vector fields (since some of these fields can be transformed to the Abel equation above), there is no doubt that the problem is important and interesting also in its own right.
The polynomials \(p\) and \(q\) are said to satisfy the Polynomial Composition Conjecture (PCC) on \(A\) if there exist polynomials \(\widetilde P\), \(\widetilde Q\) and \(W\) such that \(P=\int p\) and \(Q=\int q\) are representable as \( P(x)=\widetilde P(W(x))\), \(Q(x)=\widetilde Q(W(x)), \) and \( W(a_1)=\dots=W(a_r)\).
The PCC is a sufficient condition for the center problem and it is known also to be necessary for the center problem for small degrees of \(p\) and \(q\) and in some other very special situations.
The authors show that, for wide ranges of degrees of \(P\) and \(Q\) (restricted only by certain assumptions on the common divisors of their degrees), the composition condition provides a very accurate approximation of the center one – up to a finite number of configurations not accounted for. It appears to be the first general (that is, not restricted to small degrees of \(p\) and \(q\) or to a very special form of these polynomials) result regarding the center problem for Abel equations.
As an important intermediate result, it is shown that “at infinity” (according to an appropriate projectivization of the parameter space) the center conditions are given by a system of the moment equations of the form \(\int^{a_s}_{a_1} P^k q = 0\), \(s=2,\dots,r\), \(k=0,1,\dots\)


34C05 Topological structure of integral curves, singular points, limit cycles of ordinary differential equations
34C07 Theory of limit cycles of polynomial and analytic vector fields (existence, uniqueness, bounds, Hilbert’s 16th problem and ramifications) for ordinary differential equations
34C08 Ordinary differential equations and connections with real algebraic geometry (fewnomials, desingularization, zeros of abelian integrals, etc.)
34C25 Periodic solutions to ordinary differential equations
30E05 Moment problems and interpolation problems in the complex plane
34M35 Singularities, monodromy and local behavior of solutions to ordinary differential equations in the complex domain, normal forms
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