Algebraic geometry of Abel differential equation
type: Item
instance of: scholarly article
title: Algebraic geometry of Abel differential equation
zbMATH Open document ID: 1308.34037
author: Sh. Giat
author: Yonatan Shelah
author: Clara Shikhelman
author: Yosef Yomdin
published in: Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A: Matemáticas. RACSAM
publication date: 2014-09-18T00:00:00Z
review text: Consider a system of differential equations \[ \dot{x}= -y + F(x,y), \qquad \dot{y}=x+G(x,y), \tag{\(*\)} \] where \(F\) and \(G\) are analytic functions without constant and linear terms. This system has a center at the origin if all the solutions around the origin are periodic. The classical center-focus problem consists in finding necessary and sufficient conditions on \(F\) and \(G\) to have a center at the origin. Center conditions are given by an infinite system of polynomial equations in the coefficients of \(F\) and \(G\) and the ideal generated by these equations determines the center variety. Hence, the algebraic geometry enters to this problem in a natural way. If \(F\) and \(G\) are homogeneous polynomials then system \((*)\) can be reduced by a Cherkas transformation to the Abel equation \[ \frac{d \rho}{ d \theta} = p(\theta) \rho^2 + q(\theta) \rho^3, \tag{\(**\)} \] where \(\theta \in [0, 2 \pi]\) and \(p\) and \(q\) are trigonometric polynomials. Under this tranformation periodic solutions of system \((*)\) become periodic solutions of \((**)\). Hence, a natural extended setting for the center-focus problem is to consider a general (not necessary polynomial) Abel equation \[ y'= p(x) y^2 + q(x) y^3, \] with \(x \in [a,b]\) and repalacing the periodicity of the solutions by the condition \(y(a)=y(b)\). In this case the center conditions are expressed in a rather explicit form as a linear combinations of iterated integrals of \(p\) and \(q\). Recently two important structures have been related to the center equations for equation \((**)\): composition algebra and moment vanishing. In the present paper the authors give an overview of the relations between the center set, composition set and moment set from the results obtained in previous works, stressing their algebraic-geometric interpretation and consequences. The second part of the paper is devoted to a rather detailed study of a specific example of the Abel equation which possesses algebraic solut…
reviewed by: Jaume Giné
Mathematics Subject Classification ID: 34C05
Mathematics Subject Classification ID: 34A34
Mathematics Subject Classification ID: 34C07
Mathematics Subject Classification ID: 14H70
zbMATH DE Number: 6345557
zbMATH Keywords: Abel equations
zbMATH Keywords: center problem
zbMATH Keywords: composition condition
zbMATH Keywords: moment problem
Wikidata QID: Q115375824
MaRDI profile type: Publication
full work available at URL: s13398-012-0112-4
OpenAlex ID: W1971638361
cites work: Non-autonomous equations related to polynomial two-dimensional systems
cites work: Center conditions, compositions of polynomials and moments on algebraic curves
cites work: Parametric center-focus problem for Abel equation
cites work: Center conditions at infinity for Abel differential equations
cites work: On the center problem for ordinary differential equations
cites work: Center Problem for the Group of Rectangular Paths
cites work: Abel ODEs: Equivalence and integrable classes
cites work: Limit cycles of differential equations
cites work: Periodic orbits in complex Abel equations
cites work: Centers for trigonometric Abel equations
cites work: THE GROUP OF TRANSLATIONS AND POSITIVE RATIONAL POWERS IS FREE
cites work: Analytic continuation and fixed points of the Poincaré mapping for a polynomial Abel equation
cites work: Limit Cycles for a Class of Abel Equations
cites work: Centennial History of Hilbert's 16th Problem
cites work: On the number of solutions of the equation \(\sum^n_{j=0}a_j(t)x^j,0\leq t\leq 1\), for which \(x(0)=x(1)\)
cites work: The Number of Periodic Solutions of the Equation Ż=z N +p 1 (t )z
cites work: Q5199458
cites work: Generalized ‘second Ritt theorem’ and explicit solution of the polynomial moment problem
cites work: On polynomials orthogonal to all powers of a given polynomial on a segment
cites work: Solution of the polynomial moment problem
DOI: 10.1007/S13398-012-0112-4
description: scientific article; zbMATH DE number 6345557