Center conditions at infinity for Abel differential equations
type: Item
instance of: scholarly article
title: Center conditions at infinity for Abel differential equations
zbMATH Open document ID: 1216.34025
author: M. Briskin
author: N. Roytvarf
author: Yosef Yomdin
published in: Annals of Mathematics. Second Series
publication date: 2010-09-06T00:00:00Z
review text: 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…
Mathematics Subject Classification ID: 34C05
Mathematics Subject Classification ID: 34C07
Mathematics Subject Classification ID: 34C08
Mathematics Subject Classification ID: 34C25
Mathematics Subject Classification ID: 30E05
Mathematics Subject Classification ID: 34M35
zbMATH DE Number: 5779361
zbMATH Keywords: center-focus problem
zbMATH Keywords: Abel differential equation
zbMATH Keywords: polynomial composition conjecture
zbMATH Keywords: generalized moment problem
MaRDI profile type: Publication
full work available at URL: annals.2010.172.437
OpenAlex ID: W2010471949
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Center conditions at infinity for Abel differential equations (DOI)
DOI: 10.4007/ANNALS.2010.172.437

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