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Center and Composition Conditions for Abel Equation

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Center and Composition Conditions for Abel Equation Center and Comp osition conditions for Ab el Equation Thesis for the degree of Do ctor of Philosophy by Michael Blinov Under the Sup ervision of professor Yosef Yomdin Department of Theoretical Mathematics The Weizmann Institute of Science Submitted to the Feinb erg Graduate Scho ol of the Weizmann Institute of Science Rehovot Israel June Acknowledgments I would like to thank my advisor Professor Yosef Yomdin for his great sup ervision constant supp ort and encouragement for guiding and stimulat ing my mathematical work I should also mention that my trips to scien tic meetings were supp orted from Y Yomdins Minerva Science Foundation grant I am grateful to JP Francoise F Pakovich E Shulman S Yakovenko and V Katsnelson for interesting discussions I thank Carla Scapinello Uruguay Jonatan Gutman Israel Michael Kiermaier Germany and Oleg Lavrovsky Canada for their results obtained during summer pro jects that I sup ervised I am also grateful to my mother for her patience supp ort and lo ve I wish to thank my mathematical teacher V Sap ozhnikov who gave the initial push to my mathematical research Finally I wish to thank the following for their friendship and in some cases love Lili Eugene Bom Elad Younghee Ira Rimma Yulia Felix Eduard Lena Dima Oksana Olga Alex and many more Abstract We consider the real vector eld f x y gx y on the real plane IR This vector eld describ es the dynamical system x f x y y g x y A classical CenterFo cus problem whichwas stated byHPoincare in s is nd conditions on f g under which all tra jectories of this dynamical sys tem are closed curves around some point This situation is called a center In some cases this problem is reduced to the problem of nding conditions for solutions of Ab el dierential equation p q to be p erio dic on the interval In the thesis various sp ecial cases of the CenterFocus problem for Ab el Equation are investigated through their relation to Comp osition algebra of p olynomials and rational functions to Generalized Moments and to Algebraic Geometry of Plane Curves Contents Intro duction Center and Comp osition Conditions for vector elds on the plane Intro duction Cherkas transform General transform from a plane vector eld to a rst order nonautonomous dierential equation Comp osition conditions for Symmetric and Hamiltonian com ponents in general case Hamiltonian system Symmetric comp onent Comp osition and center conditions for quadratic planar systems The comp osition condition for the Hamiltonian com p onent Comp osition condition for the symmetric comp onent Noncomp osition comp onents for quadratic systems Cubic p olynomial vector elds Comp osition conditions for the Hamiltonian comp o H nent C condition for the comp onent C do es Comp osition not hold Comp osition conditions for the Symmetric comp onent R C Poincare return map for Ab el dierential equation and com putations of center conditions Intro duction Poincare return map for Ab el equation and recurrence relations Generators of ideals I k Center and Comp osition Conditions for Ab el dierential equa tions with p q trigonometric p olynomials of small degrees Intro duction Computational approach Center conditions for p q of small degrees Center conditions for the case when either p or q is of the degree zero Center conditions for the case deg p deg q Center conditions for the case deg p deg q Center conditions for the case deg p deg q Center conditions for the case deg p deg q Center and Comp osition Conditions for Ab el dierential equa tions with p q algebraic p olynomials of small degrees Intro duction Comp osition conjecture for multiple zero es and its status for small degrees of p and q Description of a center set for p q of small degrees Comp osition conjecture for some sp ecial families of p olynomials Riemann Surface approach to the Center problem for Ab el equation Intro duction Ab el equation on Riemann Surfaces Example Polynomial Comp osition Conjecture in the case X C P and Q p olynomials in x Example Laurent Comp osition Condition in the case X a neighb orho o d of the unit circle S fjxj g C P and Q Laurent series convergent on X Poincare mapping Factorization comp osition for the case X C P and Q rational functions Intro duction Comp osition and rational curves Structure of comp osition in the case X C P and Q poly nomials Structure of comp osition in the case X a neighborhood of the unit circle on C P and Q Laurent p olynomials Moments of P Q on S and Center Conditions for Ab el equa tion with rational p q Intro duction Sucient center condition for Ab el equation with analytic p q The degree of a rational mapping and an image of a circle on a rational curve Center conditions for the case of P Q Laurent p olynomials Generalized Moments and Center Conditions for Ab el dif ferential equation with elliptic p q Intro duction Generalized Moments on Elliptic Curves Ab el equation with co ecientselliptic functions Chapter Intro duction Let F x y Gx y b e algebraic p olynomials in x y without free and linear terms Consider the system of dierential equations x y F x y y x Gx y One can reduce the system with homogeneous F G of degree d to Ab el dierential equation p q where p q are p olynomials in sin cos of degrees dep ending only on d Then has a center if and only if has all the solutions p erio dic satisfying y y on ie solutions The classical center problem as stated for is to nd conditions on p and q such that has a center We shall call it Center Condition The following simple sucient condition was intro duced in AL Let w C be a function such that w w Let p pw w q qw w Then all the solutions of have the form w hence they satisfy the condition We shall call the representation Comp osition Condition on p q The comp osition condition is sucient but not necessary for the center The main sub ject of this research is the investigation of relations b etween center and comp osition conditions In the next chapter I intro duce statement of center problem for vector elds on the plane Reduction of the problem to the problem for Ab el dif ferential equation Cherkas transform is demonstrated and an attempt is done to generalize it Center and comp osition conditions are investigated for dynamical systems on the plane with F G homogeneous p olynomials of degrees and We show which center comp onents can b e represented as a comp osition and which can not In the third chapter I intro duce Poincare return map for Ab el dierential equation Recurrence relations and corresp onding ideals are studied and generators for these ideals are found In the fourth chapter of the thesis Ab el dierential equation is studied with p q trigonometric p olynomials of small degrees up to It is shown that in these cases center and comp osition conditions coincide although it is known that for greater degrees of p q some center conditions are not given by comp osition In the fth chapter of the thesis center and comp osition conditions are studied for Ab el dierential equation with p q algebraic p olynomials of small degrees It was shown that these conditions coincide Some generaliza tion of center problem are intro duced Weshow also some sp ecial families of p olynomials for which equivalence of the center and the comp osition condi tions can be easily sho wn In the sixth chapter Riemann Surface approachtotheCenter problem for Ab el equation is discussed This is a convenient general setting where Ab el Equation is considered on a given Riemann Surface The notions of center and comp osition are generalized accordingly In the seventh chapter we study comp osition for rational functions We establish some facts ab out structure of comp osition for algebraic p olynomials and Laurent p olynomials In the eighth chapter center problem is studied for Ab el dierential equa tion with rational functions p q One can rewrite the dierential equation i in a complex form expressing sin and cos through z e ie z z cos z z sin i y so one obtains p and q in the form of Laurent p olynomials in z and Ab el dierential equation is dy pz y q z y dz considered on the circle z S It is shown that certain integral con R R dition on P p and Q q namely vanishing of generalized moments R i j P Q dP implies center jz j Finally in the last chapter of the thesis Riemannsurface approach is extended for the case of elliptic functions Sp ecically for elliptic functions R i j P Q dP P Q it is shown that in general the condition do es not jz j imply center The dierence between
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