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Page 1: University of Connecticut · Ac kno wledgmen ts I w ould lik e to thank m y advisor Professor Y osef omdin for his great sup ervision constan t supp ort and encouragemen t for guiding

Center and Composition conditions for Abel

Equation

Thesis for the degree of Doctor of Philosophy

by

Michael Blinov

Under the Supervision of professorYosef Yomdin

Department of Theoretical MathematicsThe Weizmann Institute of Science

Submitted to the Feinberg Graduate School ofthe Weizmann Institute of Science

Rehovot ������ Israel

June ��� ����

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Acknowledgments

I would like to thank my advisor� Professor Yosef Yomdin� for his greatsupervision� constant support and encouragement� for guiding and stimulat�ing my mathematical work� I should also mention that my trips to scien�tic meetings were supported from Y� Yomdins Minerva Science Foundationgrant�I am grateful to J��P� 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 obtainedduring summer projects that I supervised�I am also grateful to my mother for her patience� support and love� I wish

to thank my mathematical teacher V� Sapozhnikov� who gave the initial pushto 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�

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Abstract

We consider the real vector eld �f�x� y�� g�x� y�� on the real plane IR��This vector eld describes the dynamical system�

�x � f�x� y��y � g�x� y�

A classical Center�Focus problem� which was stated by H� Poincare in ����s�is nd conditions on f � g� under which all trajectories 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 conditionsfor solutions of Abel di�erential equation

�� � p����� � q�����

to be periodic on the interval ��� ����In the thesis various special cases of the Center�Focus problem for Abel

Equation are investigated� through their relation to Composition algebra ofpolynomials and rational functions� to Generalized Moments and to AlgebraicGeometry of Plane Curves�

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Contents

� Introduction �

� Center and Composition Conditions for vector �elds on theplane ���� Introduction � � � � � � � � � � � � � � � � � � � � � � � � � � � � ���� Cherkas transform � � � � � � � � � � � � � � � � � � � � � � � � ����� General transform from a plane vector eld to a rst order

non�autonomous di�erential equation � � � � � � � � � � � � � � ����� Composition conditions for Symmetric and Hamiltonian com�

ponents in general case � � � � � � � � � � � � � � � � � � � � � � ������� Hamiltonian system � � � � � � � � � � � � � � � � � � � ������� Symmetric component � � � � � � � � � � � � � � � � � � ��

��� Composition and center conditions for quadratic planar systems ������� The composition condition for the Hamiltonian com�

ponent � � � � � � � � � � � � � � � � � � � � � � � � � � � ������� Composition condition for the symmetric component � ������� Non�composition components for quadratic systems � � ��

��� Cubic polynomial vector elds � � � � � � � � � � � � � � � � � � ������� Composition conditions for the Hamiltonian compo�

nent �CH� � � � � � � � � � � � � � � � � � � � � � � � � � � ��

����� Composition condition for the component �C�� doesnot hold � � � � � � � � � � � � � � � � � � � � � � � � � � ��

����� Composition conditions for the Symmetric component�CR

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ��

� Poincare return map for Abel di�erential equation and com�putations of center conditions ��

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��� Introduction � � � � � � � � � � � � � � � � � � � � � � � � � � � � ����� Poincare return map for Abel equation and recurrence relations ����� Generators of ideals Ik � � � � � � � � � � � � � � � � � � � � � � ��

Center and Composition Conditions for Abel di�erential equa�tions with p q � trigonometric polynomials of small degrees ����� Introduction � � � � � � � � � � � � � � � � � � � � � � � � � � � � ����� Computational approach � � � � � � � � � � � � � � � � � � � � � ����� Center conditions for p� q of small degrees � � � � � � � � � � � ��

����� Center conditions for the case when either p or q is ofthe 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 Composition Conditions for Abel di�erential equa�tions with p q � algebraic polynomials of small degrees ��� Introduction � � � � � � � � � � � � � � � � � � � � � � � � � � � � ����� Composition conjecture for multiple zeroes and its status for

small degrees of p and q � � � � � � � � � � � � � � � � � � � � � ����� Description of a center set for p� q of small degrees� � � � � � � ����� Composition conjecture for some special families of polynomials ��

� Riemann Surface approach to the Center problem for Abelequation ����� Introduction � � � � � � � � � � � � � � � � � � � � � � � � � � � � ����� Abel equation on Riemann Surfaces � � � � � � � � � � � � � � ��

����� Example Polynomial Composition Conjecture in thecase X � C � P and Q � polynomials in x � � � � � � � � ��

����� Example Laurent Composition Condition in the caseX � a neighborhood of the unit circle S� � fjxj � �g �C � P and Q � Laurent series� convergent on X � � � � � ��

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��� Poincare mapping � � � � � � � � � � � � � � � � � � � � � � � � ��

� Factorization composition� for the case X � C P and Q �rational functions ����� Introduction � � � � � � � � � � � � � � � � � � � � � � � � � � � � ����� Composition and rational curves � � � � � � � � � � � � � � � � � ����� Structure of composition in the case X � C � P and Q � poly�

nomials � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ����� Structure of composition in the case X � a neighborhood of

the unit circle on C � P and Q � Laurent polynomials � � � � � ��

� Moments of P Q on S� and Center Conditions for Abel equa�tion with rational p q ����� Introduction � � � � � � � � � � � � � � � � � � � � � � � � � � � � ����� Su�cient center condition for Abel 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 polynomials � ��

� Generalized Moments and Center Conditions for Abel dif�ferential equation with elliptic p q ����� Introduction � � � � � � � � � � � � � � � � � � � � � � � � � � � � ����� Generalized Moments on Elliptic Curves � � � � � � � � � � � � ����� Abel equation with coe�cients�elliptic functions � � � � � � � � ��

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Chapter �

Introduction

Let F �x� y�� G�x� y� be algebraic polynomials in x� y without free and linearterms� Consider the system of di�erential equations�

�x � �y � F �x� y��y � x�G�x� y�

�����

One can reduce the system ����� with homogeneous F � G of degree d to Abeldi�erential equation

�� � p����� � q����� �����

where p���� q��� are polynomials in sin �� cos � of degrees depending only ond� Then ����� has a center if and only if ����� has all the solutions periodicon ��� ���� i�e� solutions � � ���� satisfying y��� � y�����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 su�cient condition was introduced in �AL�� Let

w��� � C���� ��� be a function such that w��� � w����� Let�p��� � �p�w����w����q��� � �q�w����w�����

�����

Then all the solutions of ����� have the form ���� � ���w����� hence theysatisfy the condition ���� � ������We shall call the representation ����� Composition Condition on p� q�

The composition condition is su�cient� but not necessary for the center�

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The main subject of this research is the investigation of relations betweencenter and composition conditions�In the next chapter I introduce statement of center problem for vector

elds on the plane� Reduction of the problem to the problem for Abel dif�ferential equation �Cherkas transform� is demonstrated and an attempt isdone to generalize it� Center and composition conditions are investigated fordynamical systems on the plane with F � G � homogeneous polynomials ofdegrees � and �� We show which center components can be represented as acomposition and which can not�In the third chapter I introduce Poincare return map for Abel di�erential

equation� Recurrence relations and corresponding ideals are studied andgenerators for these ideals are found�In the fourth chapter of the thesis Abel di�erential equation is studied

with p� q � trigonometric polynomials of small degrees �up to ��� It is shownthat in these cases center and composition conditions coincide� although it isknown that for greater degrees of p� q some center conditions are not givenby composition�In the fth chapter of the thesis center and composition conditions are

studied for Abel di�erential equation with p� q � algebraic polynomials ofsmall degrees� It was shown that these conditions coincide� Some generaliza�tion of center problem are introduced� We show also some special families ofpolynomials� for which equivalence of the center and the composition condi�tions can be easily shown�In the sixth chapter Riemann Surface approach to the Center problem for

Abel equation is discussed� This is a convenient general setting� where AbelEquation is considered on a given Riemann Surface� The notions of centerand composition are generalized accordingly�In the seventh chapter we study composition for rational functions� We

establish some facts about structure of composition for algebraic polynomialsand Laurent polynomials�In the eighth chapter center problem is studied for Abel di�erential equa�

tion with rational functions p� q� One can rewrite the di�erential equation����� in a complex form� expressing sin � and cos � through z � ei�� i�e���

�cos � � z�z��

��

sin � � z�z��

�i�

� � y�

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so one obtains p and q in the form of Laurent polynomials in z� and Abeldi�erential equation is

dy

dz� p�z� y� � q�z� y�

considered on the circle z���� ���� � S�� It is shown that certain integral con�dition on P �

Rp and Q �

Rq �namely vanishing of generalized momentsR

jzj��P iQjdP � �� implies center�

Finally� in the last chapter of the thesis Riemann�surface approach isextended for the case of elliptic functions� Specically� for elliptic functionsP � Q it is shown that in general the condition

Rjzj��

P iQjdP � � does not

imply center� The di�erence between the Laurent polynomials and ellipticfunctions P � Q is shown to be in the topology of the complex curve Y �f�P �z�� Q�z�� z � C g � C

��� In the rst case this curve is rational� and inthe second case it is topologically a torus� The one�dimensional homology ofthe torus is responsible for ramication of the solutions of the Abel equation�in spite of vanishing of all the moments

Rjzj��

P iQjdP �

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Chapter �

Center and CompositionConditions for vector �elds onthe plane

��� Introduction

The following formulation of the center problem �see e�g� �Sch� for a generaldiscussion of the classical center problem� is considered Let F �x� y�� G�x� y�be algebraic homogeneous polynomials in x� y of degree d� Consider thesystem of di�erential equations�

�x � �y � F �x� y��y � x�G�x� y�

�����

A solution x�t�� y�t� of ����� is said to be closed if it is dened in theinterval ��� t�� and x��� � x�t��� y��� � y�t��� The system ����� is said tohave a center at � if all the solutions around zero are closed� Then thegeneral problem is under what conditions on F�G the system ����� has acenter at zero�It was shown in �Ch� that one can reduce the system ����� with homoge�

neous F � G of degree d to Abel equation

�� � p����� � q����� �����

where p���� q��� are polynomials in sin�� cos� of degrees d� � and �d� �

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respectively� Then ����� has a center if and only if ����� has all the solutionsperiodic on ��� ���� i�e� all the solutions r � r��� with ���� su�ciently smallsatisfy ���� � ������ Conditions on coe�cients of p� q� under which ����� hasall the solutions periodic� are called center conditions for Abel equation������

Following �AL�� in �BFY��BFY�� the following simple su�cient condition

was stated� Let P ��� �

Z �

p�s�ds� Q��� �

Z �

q�s�ds� Then

Composition Condition� The su�cient condition for the existence of thecenter for ����� is the representability of P ���� Q��� as a composition of al�gebraic polynomials with trigonometric polynomial� i�e� there exist a trigono�metric polynomial W ��� � polynomial of cos �� sin � s�t� W ��� � W ���� ��� and algebraic polynomials �P � �Q without free terms� s�t� P ��� � �P �W �����Q��� � �Q�W �����

The su�ciency follows from the fact� that if P � �P �W �� Q � �Q�W �� thenthe solution of ����� ���� � ���W �� and as W ��� � W ����� ���� � ������The full center conditions are known yet only for homogeneous perturba�

tions F � G of degrees d � �� �� In this chapter we

� demonstrate Cherkas transform using normal form technique �sections���������

� show following �AL� that Hamiltonian and Symmetric components aregiven in fact by composition �section �����

� explicitly write down composition representation for Hamiltonian andSymmetric components in quadratic �section ���� and cubic �section���� cases�

� show by counterexample that other known components for quadraticand cubic homogeneous vector elds are in general unrepresentable asa composition�

��

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��� Cherkas transform

We shall show how in a special case of systems ������ the problem of ndingcenter conditions on F �x� y�� G�x� y� can be reduced to the center problemfor Abel equation ������Write ����� in polar coordinates x � r cos �� y � r sin �� We obtain�

�x � �r cos � � r sin � � �� � �r sin � � F �r cos �� r sin ���y � �r sin � � r cos � � �� � r cos � � F �r cos �� r sin ��

Multiplying the rst equation by cos �� the second by sin � and adding�we get

�r � F �r cos �� r sin �� cos � �G�r cos �� r sin �� sin �

� rd�F �cos �� sin �� cos � �G�cos �� sin �� sin �� � rdf���

In a similar way�

�� � � � rd����F �cos �� sin �� sin � �G�cos �� sin �� cos �� � � � rd��g���

Finally� we getdr

d���r���

rdf���

� � rd��g���

Then we apply a transformation� suggested by L� Cherkas in �Che�� ������ �

rd��

� � rd��g���

rd�� ��

�� �g���

Notice that for d � � this transformation and its inverse are regular atr � � � �� For d � �� Cherkas transformation� considered over complexnumbers� ramies at zero� but for small real r � �� � � �� it is monotone�Denote � � rd��g��� by A� We have

d�

d���

A�

��d� ��rd�� � dr

d�� A� rd��

��d� ��rd��dr

d�g��� � rd��g����

��

�rd��

A�

���d� ��f��� � g����� rd�� � �d� ��f���g���r

�d��

A

��

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� ��d� ��f��� � g����� �� � �d� ��f���g�����

Thus we obtaind�

d�� p����� � q�����

where p��� � �d���f����g���� and q��� � ��d���f���g��� are polynomialsin sin �� cos � of degrees d� � and �d� �� respectively�Clearly� the trajectory of ����� near the origin is closed if and only if the

corresponding solution of ����� satises ����� � ���� �and hence is periodic��Therefore the center�focus problem for ����� is translated into the problemof nding conditions on p and q� under which all the solutions of ����� areperiodic�

��� General transform from a plane vector

�eld to a �rst order non�autonomous dif�

ferential equation

In this section a transform from a polynomial plane vector eld to a rst ordernon�autonomous di�erential equation is illustrated� Let F �x� y�� G�x� y� bearbitrary polynomials of degree grater than �� We can consider any polyno�mials F �x� y�� G�x�y� as a sum of homogeneous polynomials Fd� Gd of degreed �

F � F� � F� � � � �� FkG � G� �G� � � � ��Gk

In a similar way to the homogeneous case we obtain

dr

d���r���

r�f���� � r�f���� � � � �� rkfk���

� � rg���� � r�g���� � � � �� rk��gk���

When r is su�ciently small� we can write it as

dr

d���r�f���� � r�f���� � � � �� rkfk���

� �Xi��

�rg�����r�g������ � ��r

k��gk����i

��

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After multiplication we obtain formal innite power series

�r ��Xi��

ai ri�

where ai are polynomials in sin �� cos ��Now we can apply normal form technique� eliminating one by one all the

terms of degrees higher than �� Namely� substituting u � r � ��t� r�� we get

�u � �r��� r���� r� �r � �r ����� r����� r� � ��Xi��

ai ri������ �

�Xi��

ai ri������ �u�� u���

� a�u� � a�u

� � �a� � ��u� � ����

So by choosing �� � �a� we get di�erential equation with the right handside � power series without u�� Continuing elimination� we can kill all termsuk for k � � �there are some obstructions to this process� which we dontdiscuss now�� These initial computations suggest a way to transform anypolynomial vector eld to a rst order non�autonomous di�erential equation�

��� Composition conditions for Symmetric and

Hamiltonian components in general case

����� Hamiltonian system

We consider the system ��x � �y � �H

�y�x� y�

�y � x � �H�x�x� y�

with H�x� y� � homogeneous polynomial in x� y of degree d��� After trans�forming to polar coordinates we get�

�r � rdf����� � � � rd��g���

with

f��� � ��H�y�cos � sin �� cos � �

�H

�x�cos �� sin �� sin ��

��

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hence f��� � �dHd�H�cos �� sin ���

g��� ��H

�y�cos � sin �� sin � �

�H

�x�cos �� sin �� cos ��

Due to homogeneity of H g��� � �d� ��H�cos �� sin ���Applying Cherkas transformation� we obtain the Abel equation �� �

p����� � q������ with p��� � �d� ��f���� g����� q��� � ��d� ��f���g����Integrating� we get

P ��� � ��d���H�cos �� sin ����d���H�cos �� sin ���C� � ��dH�cos �� sin ���C��

q��� � �d� � ��dH�cos �� sin ��dt

H�cos �� sin ���

hence

Q��� �d� � ��

H��cos �� sin �� � C��

and

Q��� �d� � ��d�

P ���� � C�P ����

where C� is a constant�

����� Symmetric component

We consider the planar system��x � �y � F �x� y��y � x�G�x� y�

As we can rotate our system around the origin� it is enough to consider thecase of symmetry with respect to x�axis �

F �x��y� � �F �x� y�G�x��y� � G�x� y�

In polar coordinates we get f���� � F �cos ��� sin �� cos ��G�cos ��� sin ���� sin �� ��f���� g���� � �F �cos ��� sin ���� sin �� �G�cos ��� sin �� cos � � g����After Cherkas transformations we get p��� � �d� ��f���� g����� q��� �

��d � ��f���g���� hence p��� and q��� are odd functions� hence P ��� andQ��� are even functions� i�e� they are functions of cos ��

��

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Below we write down explicit expressions for coe�cients of such compo�sitions for Hamiltonian and Symmetric cases for homogeneous polynomialplanar systems of the second and the third degrees�

��� Composition and center conditions for quadratic

planar systems

The system ����� with homogeneous F �x� y�� G�x� y� of degree � can berewritten in the complex notations � Zoladek �Zo���� with � parameters

�z � iz � Az� �Bz�z � C�z��

where z � x� iy� A � a� � ia�� B � b� � ib�� C � c� � ic��

It is known �see �Zo���� that the components of the center set in this caseare

B � �� Lotka�Volterra component �QLV� �

Im�AB� � Im� �B�C� � Im�A�C� � �� Symmetric component �QR� �

�A� �B � �� Hamiltonian component �QH� �

A� � �B � jCj � jBj � �� Darbu component �Q��

In usual notations on the real plane we get the following system

�x � �y � �a� � b� � c��x� � ���a� � �c��xy � ��a� � b� � c��y

�y � �x � �a� � b� � c��x� � ���a� � �c��xy � ��a� � b� � c��y

The center conditions are as follow

b� � �� b� � �� �QLV� ���

�a�b� � a�b� � �b��c� � b��c� � �b�b��c� � �b��b�c� � ��a��c� � a��c� � �a�a��c� � �a��a�c� � �

� �QR� �

�a� � b� � �� �a� � b� � �� �QH� �

��

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���a� � �b� � �a� � �b� � �c�� � c�� � b�� � b��

� �Q��

We shall nd the direct representations of Symmetric and Hamiltoniancomponents as a composition� and we shall show that in general case Lotka�Volterra and Darbu components can not be represented as a composition�

����� The composition condition for the Hamiltonian

component

Lemma� In Hamiltonian case the system has the form

�x � �y � ��a� � c��x� � ���a� � �c��xy � ���a� � c��y

�y � �x � ��a� � c��x� � ���a� � �c��xy � �a� � c��y

and there exists a composition of the form

Q�x� ��

���P ��x� �

���a� � c��P �x��

Proof� Using the Mathematica program �here is the main function�

F��x��y�����lambda� x� lambda� y��

G��x��y���� � lambda� lambda�� x y �

f�G��Cos�fi��Sin�fi���Sin�fi�F��Cos�fi��Sin�fi���Cos�fi��

g�G��Cos�fi��Sin�fi���Cos�fi��F��Cos�fi��Sin�fi���Sin�fi��

pa�TrigExpand�f�D�g�fi���

qa�TrigExpand��f�g��

P�TrigReduce�Integrate�pa��fi���x����

Q�TrigReduce�Integrate�qa��fi���x����

we nd

P � ��������a�� c���a� cos�x�� c� cos��x�� �a� sin�x�� c� sin��x���

Q � �������a�� � �a�� � c�� � ��a�c� � c�� � �a�� cos��x� � �a�� cos��x� ��a�c� cos��x� � �a�c� cos��x� � �a�c� cos��x� � �a�c� cos��x� � c�� cos��x� �

��

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c�� cos��x� � ��a�a� sin��x� � �a�c� sin��x� � �a�c� sin��x� � �a�c� sin��x� ��a�c� sin��x�� �c�c� sin��x���

Now to choose in the representation Q�x� � ����P ��x� � P �x�� we

compute P ��� � ������a�� c��� Q��� � �� hence � �������a�� c��� andthe identity

Q�x� ��

���P ��x� �

���a� � c��P �x��

holds� �

����� Composition condition for the symmetric com�

ponent

Lemma� In Symmetric case there exists a composition with the polynomialof the �rst degree of the formW �x� � cos x�� sinx��� where � is a constantwhich is found from the cubic equation involving coe�cients of the system�

Proof� In this case the center conditions are���a�b� � a�b� � �b��c� � b��c� � �b�b��c� � �b��b�c� � ��a��c� � a��c� � �a�a��c� � �a��a�c� � �

Without loss of generality we may assume that b� �� � �if both b� and b�are zeroes� we are in Lotka�Volterra case�� Then from the rst equation

a� � �a�b�b��and from the two last equation we can nd either

�b�� � b��� c� � �

or

�b�� �� b��� c� ��

�b�c��

b�� �b��b�� � b��

��

Consider the rst case� when b� � b�p�� Running Mathematica� we get

P � �p�b� � �

p�b� cos�x� � �b� sin�x� � ��c� sin��x��

��

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Q � �������a��� �b��� �p�a�c���b�c�� c��� �a�� cos��x���b�� cos��x��

�p�a�c� cos��x���b�c� cos��x��c�� cos��x���

p�a�� sin��x���

p�b�� sin��x��

�a�c� sin��x�� �p�b�c� sin��x���

After some computations we obtain composition with the rst degree

polynomial W �x� � cos�x�� �p�sin�x�� �

P �x� � ���c�p��W �x�� � ��c�

p��W �x�� � ��c�

p� � �b�

p��W �x��

Q � ��

�c���W

� ���c���

W � ����c�� � ��b�c�

�W � �

��b�c� � ��c��

�W �

��a�� � �b�� � a�c�

p� � ��b�c� � �c�

�W � � ��a�� � �b�� �

p�a�c� � �b�c��W�

Other two cases were considered similarly� and in each of them we gotcomposition� �

����� Non�composition components for quadratic sys�

tems

One of ways to check if two polynomials P �x�� Q�x� have a common divisoris to plot graphs of polynomials P �x� � P �y� and Q�x� � Q�y� on the realplane �x� y� and to see if there is a common subgraph outside the diagonalx � y� For real curves these subgraphs are real� �see �ER� for detail��We plot graphs for di�erent numerical values in each of the four compo�

nents � using the �Matlab� software�For Lotka�Volterra component we consider the numerical example with

parameters a� � �� a� � �� c� � �� c� � �� i�e��F �x� y� � �x� � �y�G�x� y� � �x� � �xy � �y� �

For Darbu component we consider the numerical example with parame�ters a� � �� a� � �� c� � �� c� � �� i�e��

F �x� y� � �x� � �y�G�x� y� � �x� � �xy � �y� �

��

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−3 −2 −1 0 1 2 3−3

−2

−1

0

1

2

3

Plots of P(x)−P(y) (red) and Q(x)−Q(y) (green).

XY

Figure ��� Graph for Lotka�Volterra component

−3 −2 −1 0 1 2 3

−3

−2

−1

0

1

2

3

Plots of P(x)−P(y) (red) and Q(x)−Q(y) (green).

X

Y

Figure ��� Graph for Darbu component

For Lotka�Voltera and Darbu components we see that there is no commonsubgraph� Although rigorous proof would require discussion of computer pre�cision of Matlab software and graphical capabilities of hardware� we assumethat visually observable distance between curves is greater than machine er�ror� and there is no common subgraph� These graphical counterexamplesprove that there is no composition for these two components�

��� Cubic polynomial vector �elds

The system ����� with homogeneous F �x� y�� G�x� y� of degree � can berewritten in the complex notations � Zoladek �Zo��� with � parameters

�z � iz �Dz� � Ez��z � Fz�z� �G�z��

where z � x� iy� D � d� � id�� E � e� � ie�� F � f� � if�� G � g� � ig��

��

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It is known �see �Zo���� that the components of the center set in this caseare

Re�E� � �D � �F � �� �CH� �

Re�E� � Im�DF � � Re�D�G� � Re� �F �G� � �� �CR� �

E � D � � �F � jGj � �jF j � �� �C��

In usual notations on R� we get the following system

�x � �y � �d� � e� � f� � g��x� � ���d� � e� � f� � �g��x

�y

��e� � �d� � f� � �g��xy� � �d� � e� � f� � g��y�

�y � x� �d� � e� � f� � g��x� � ��d� � e� � f� � �g��x�y

��e� � �d� � f� � �g��xy� � ��d� � e� � f� � g��y�

The center conditions are as follow

e� � �� �d� � f� � �� �d� � f� � � �CH� ���

�e� � �� d�f� � d�f� � ��d��g� � d��g� � �d�d�g� � ��f �� g� � f �� g� � �f�f�g� � ��

�CR� �

e� � �� e� � ��d� � �f� � �� d� � �f� � ��g�� � g�� � ��f

�� � f �� �

�C��

We shall nd the direct representations of Symmetric and Hamiltoniancomponents as a composition� and we shall show that in general case Darbucomponent can not be represented as a composition�

����� Composition conditions for the Hamiltonian com�

ponent �CH��

Lemma� In Hamiltonian case the system has the form

�x � �y � ���d� � g��x� � ��e� � �g��x�y

����d� � �g��xy� � ��d� � e� � g��y�

�y � x� ��d� � e� � g��x� � ��d� � �g��x�y

��e� � �g��xy� � ��d� � g��y�

��

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and there exists a composition of the form

Q��� ��

��P ����� ���d� � e� � g��P �����

Proof� After running Mathematica we get

P � ��d����g�����d� cos��x����g� cos��x���d� sin��x����g� sin��x���

Q � �������d�����d�����d�e��g�����d�g���e�g��g�����������d�e� cos��x���d�g� cos��x� � �d�g� cos��x�� ��d�� cos��x� � ��d�� cos��x� � �e�g� cos��x� ��d�g� cos��x���d�g� cos��x��g�� cos��x��g�� cos��x����d�e� sin��x���d�g� sin��x���d�g� sin��x����d�d� sin��x���e�g� sin��x���d�g� sin��x���d�g� sin��x���g�g� sin��x���

Substituting x � �� we get P � ��d�� Q � ��d��e� � g��� hence � �����d� � e� � g��� Now we get

Q��� � �

��P ����� ���d� � e� � g��P �����

q�e�d�So� in Hamiltonian case both P and Q can be expressed as a composition

of some usual polynomials with a certain trigonometric polynomial� namelyP � �

����� Composition condition for the component �C��does not hold

In usual notations on R� this component is given by the following system

�x � �y � ��f� � g��x� � ���f� � �g��x

�y

����f� � �g��xy� � ���f� � g��y�

�y � x � ���f� � g��x� � ��f� � �g��x�y

����f� � �g��xy� � ���f� � g��y�

with an additional relation on coe�cients g�� � g�� � ��f�� � f �� ���

��

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We compute� using program� which utilizes only the rst four of the �equations� We get

P � �f����g�����f� cos��x����g� cos��x���f� sin��x����g� sin��x���

Q � ���������f �����f �����f�g���g�����f�g���g�������������f�g� cos��x����f�g� cos��x����f

�� cos��x����f �� cos��x���f�g� cos��x���f�g� cos��x��

�g�� cos��x�� �g�� cos��x�� ��f�g� sin��x�� ��f�g� sin��x� � ��f�f� sin��x��

�f�g� sin��x�� �f�g� sin��x�� �g�g� sin��x���

where there is one additional relation on coe�cients

g�� � g�� � ��f�� � f �� ��

After trying to get a composition with polynomials of the rst �� sinx� cos� �� second ��� sin x � � cos x � �� sin �x � � cos �x � � � �� andthe forth �simply �P � degrees� we got contradiction�

����� Composition conditions for the Symmetric com�

ponent �CR��

Lemma� In symmetric case there exists a composition either with a polyno�mial of the �rst order in sinx� cos x� or of the second order� Namely�if d� �� � or d� � d� � �� there exists a composition with polynomialW �x� � cos x � � sinx � �� where � is de�ned from a cubic equation on��if d� � � and d� �� �� there exists a composition with W � cos �x�

Proof� Here we have the only one �usable� equation e� � � and three nonlin�ear equations� But we can solve the system of center conditions� consideringseparately cases d� � � and d� �� ��First we use the only condition e� � � to compute P and Q� Then we

consider several cases��� Assume rst d� �� �� Then we can substitute

f� � �d�f�d�

��

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into the third equation� and we get instead of the third equation the followingone

f ��d���d��g� � d��g� � �d�d�g�� � ��

We see� that the third equation disappeared� and we are left with condi�tions e� � �

d� �� �d��g� � d��g� � �d�d�g� � ��

Now we again consider two cases

�a�� d� � � � g� � �� f� � ��In that case we found the composition with

W �x� � �� � cos�x� � sin�x�

P � �g�W� � ��g�W

� � ��f� � ��g��W� � �f�W

Q � g��W � �g��W

� ��

�f�g� � ��g

���W

� � ��f�g� � ��g���W

��

��

����g�����f�g���e�g���f ����d���W ��

����f�g����g�����e�g���f ����d���W ��

����g���f�g���e�g��d�g���f ���e�f��d�e���d���W �����f�g���d�g���e�f���d�e��W�b�� d� �� �� then

g� � g�d�� � d���d�d�

In this case we obtain the composition with W � cos�x� � � sin�x� � ��where � is found from the cubic equation�

�� The second case to consider is d� � �� Then we get the system�d� � �� d�f� � �� d�g� � ��f �� g� � f �� g� � �f�f�g� � ��

��

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Again there are two cases

�a� �d� � �� d� � ��f �� g� � f �� g� � �f�f�g� � ��

In this case we obtain a composition with W � cos�x� � � sin�x� � ��where � is found from the cubic equation�

�b� d� � �� f� � �� g� � ��P and Q are both functions of cos��x�� �

��

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Chapter �

Poincare return map for Abeldi�erential equation andcomputations of centerconditions

��� Introduction

We consider Abel equation

y� � p�x� y� � q�x� y� �����

with p� q � any analytic functions� We can x any pont a instead of �� andask the same question Under which conditions on p and q are all trajectories�periodic�� i�e� any trajectory starting at � with the value y� assumes thesame value y� at a�In this chapter

� as in �BFY��� we write down recurrence relations for coe�cients ofPoincare return map�

� using these recurrence relations� we compute several rst coe�cientsmanually and nd out generators of coe�cient ideals�

� we describe an algorithm to nd these generators using �Mathematica�symbolic computer system�

��

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��� Poincare return map for Abel equation

and recurrence relations

We may look for solutions of ����� in the form of Poincare return map

y�x� y�� � y� ��Xk��

vk�x� �yk� � �����

where y��� y�� � y�� � ��� �� � � �� is the �nite� set of the coe�cients of p�q� Shortly we will write vk�x��

Then y�a� � y�a� y�� � y� ��Xk��

vk�a�yk� and hence the condition y�a� �

y��� is equivalent to vk�a� � � for k � �� �� � � �

One can easily show �by substitution of the expansion ����� into the equa�tion ������ that vk�x� satisfy recurrence relations

�����������

v��x� � �v��x� � �vn��� � � and

v�n�x� � p�x�Xi�j�n

vi�x�vj�x� � q�x�X

i�j�k�n

vi�x�vj�x�vk�x�� n � ������

It was shown in �BFY�� that in fact the recurrence relations ����� canbe linearized� i�e� the same ideals Ik � fv�� v�� � � � � vkgs are generated byf���x�� � � � �k�x�g� where �k�x� satisfy linear recurrence relations

����������x� � ����x� � ��n��� � � and��n�x� � ��n� ���n���x�p�x�� �n� ���n���x�q�x�� n � �

�����

which are much more convenient then ������Direct computations �including several integrations by part� give the fol�

lowing expressions for the rst polynomials �k�x�� solving the recurrence

��

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relation ����� �Here we denote P �x� �

Z x

p�t�dt� Q�x� �

Z x

q�t�dt�

���x� � �P �x����x� � P ��x��Q�x�

���x� � �P ��x� � �P �x�Q�x��Z x

q�t�P �t�dt

���x� � P ��x�� �P ��x�Q�x��Z x

q�t�P ��t�dt

��P �x�

Z x

q�t�P �t�dt��

�Q��x�

���x� � �P ��x� � ��P ��x�Q�x� � �P �x�

Z x

q�t�P ��t�dt

��Q��x�P �x�� ��P ��x�

Z x

q�t�P �t�dt� �Q�x�

Z x

q�t�P �t�dt

�Z x

q�t�P ��t�dt��

Z x

p�t�Q��t�dt

���� � P ����� ��P ����Q����Z �

P ��t�q�t�dt

���Q���� ��P �����

Z �

q�t�P ��t�dt� ���P ����

���P ���Q����Z �

P �t�q�t�dt� �P ���

Z �

q�t�P ��t�dt

��P ���Z �

Q��t�p�t�dt� ��

Z �

P �t�q�t�dt��

���Q���� �

��

�P ����Q���� � �

Z �

P �t�p�t�Q��t�dt

and similar much more longer expression was obtained for ��x��

��

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��� Generators of ideals Ik

Consequently� we get the following set of generators for the ideals �Ik� k ��� � � � � ��

I� � fPgI� � fP�QgI� � fP�Q�

ZqPg

I� � fP�Q�Z

qP�

ZqP �g

I� � fP�Q�Z

qP�

ZqP ��

Z�qP � � �

�pQ��g

I � fP�Q�Z

qP�

ZqP ��

Z�qP � � �

�pQ���

Z�qP � � �PpQ��g

I � fP�Q�Z

qP�

ZqP ��

Z�qP � � �

�pQ���

Z�qP � � �PpQ���Z

�qP � � ��Q�p� ��P �Qq � ��P �q

ZPq�g

These generators were rst computed by Alwash and Lloyd in �AL�� us�ing the nonlinear recurrence relation ������ and recomputed now using therecurrence relation ������The problem to nd generators of ideals Ik was studied also from com�

binatorial point of view by J� Devlin ��Dev��� �Dev���� We tried to write aprogram to nd all the generators� using symbolic computations with �Math�ematica� software�The di�culty lies in that the general formula for �k��� cannot be deduced�

Respectively� we can compute �k��� using the computer� for each given p����q���� but we cannot compute the formula for a general symbolic form of p���and q���� except for the formula with �k� �� integrations for the k�th term�But using integration by parts we can reduce the complexity �the numberof integrals� in the general formula� as demonstrated in section ���� Weattempted to write a program which could compute integrals in a symbolicform using integration by parts� This work was done together with OlegLavrovsky� working on this summer project under my supervision�

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Mathematica is an integrated technical computing system that allows fornumerical and symbolic computations of high complexity� However� thereare no libraries that would facilitate manipulation of symbolic functions dueto the following two di�culties a� p���� q��� are symbolic functions� making the use of standard Integrate��and Differentiate�� functions of Mathematica inapplicable�b� an interface is necessary for allowing the user to specify the transformationor exclusion of components during the iteration� because there is a certainreasonable limit on the complexity of transformations that we dene� Un�ending switching loops often rise between two or three possible formats ofexpressions as the computer is unable to decide on a result above ������To overcome the rst di�culty� we created a new set of functions� The

function i� � is used to represent an integral �replacing the standard functionIntegrate��� and dv��� correspondingly� the default function Differentiate���We compute using recursive function

f�n����f�n�� ��n�i�Expand�p f�n������ n���i�Expand�q f�n�����

with initial conditions f��� � � � f��� � ��We start with the functions p and q as arbitrary symbols� we introduce

the complimentary set P and Q so the output f�n� will contain only p� q�P � Q and integration operation i� ��

dv�P� �� p

dv�Q� �� q

i�p� �� P

i�q� �� Q

Within the Mathematica �kernel� expressions of f�n� are traced backuntil the rst dened values �f��� and f���� and each successive expressioncalculated from therein and stored�The main set of transformations was dened in this fashion

i�x� y�� �� i�x� i�y��

dv�x� y�� �� dv�x� dv�y��

dv�x� y�� �� dv�x� y dv�y� x�

i�dv� x�����x� dv�i�x�����x�

i� p P n����� �� n��� P n��

��

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etcUsing these rules� if Mathematica encounters an expression such as

��� i�p q i�p P�������

it transforms it in the following way ��� i�p� i�q i�p P�P���������i�p� i�q� i�p P�P�����

��� PQ ���� P�P�P���

This continues until the rule is no longer applicable� To overcome thesecond di�culty we dene integration by parts as

i�p B�� �� P B � i�P �dv�B��� �� Test�

where B is any expression� Test represents a procedure to display aprompt for accepting or denying transforming of the given expression� Notethat it was critical that the ordering of these expressions for execution wascarefully determined during analysis of such problems� so that the passingof conversions should follow e�ective sequences� As we can apply many dif�ferent integration by parts �taking di�erent expressions as derivatives underintegration sign�� this step requires some complicated testing procedures�A set of functions was also added that allow the user to choose terms

to break down further� again progressing the formula into a more prefer�able form� This was based on separately transforming the chosen terms andinserting the result into the general expression� then simplifying�The developed algorithm proved to be successful in the evaluation of

�k��� in the extent of comparison with currently known expressions �up tok � ��� and had the required integral complexity in higher expressions� Forcomparison� the manual computation of ���� takes many hours� programcompute it during several minutes� The work on the algorithm is not nished�as for �k��� with larger k we have a choice of many expressions to integrateby parts� and Test procedure must be done much more complicated�

��

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Chapter �

Center and CompositionConditions for Abel di�erentialequations with p� q �trigonometric polynomials ofsmall degrees

��� Introduction

As it was shown in �Ch��� starting from homogeneous perturbation Fd�x� y��Gd�x� y� of degree d� we obtain Abel equation with p� q � homogeneouspolynomials in sinx� cos x of degrees d��� �d�� respectively� It means thatthe lowest degrees of trigonometric polynomials in ������ related to dynamicalsystem ������ are � and � respectively� The center conditions for them areknown� and they are not restricted to the composition conditions�One can consider Abel equation

y� � p�x�y� � q�x�y� �����

with p�x�� q�x� � arbitrary trigonometric polynomials� i�e� polynomials of anydegree� As before� Center Condition for Abel equation is periodicity ofsolutions y�x� between two specied points � and ��� i�e� y��� � y���� forall solutions y�x� starting at � with su�ciently small initial value y����

��

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As before� composition condition in the form P �x� � �P �W �x��� Q�x� ��Q�W �x�� for some function W �x� with W ��� �W ���� is su�cient condi�tion for the center�The question �when center and composition conditions coincide� can be

stated for any functions p� q� For p�x�� q�x� � algebraic polynomials in x ofthe form anx

n � an��xn�� � � � � � a� this question was studied in �BFY����

BY��� and the following Composition Conjecture was stated for alge�braic polynomials p� q composition condition is not only su�cient� but alsonecessary for the center� The computational verication for p� q � algebraicpolynomials of small degrees was done in �BY���We consider trigonometric polynomials in a general form

p��� � � �

deg p���Xk��

��k�� sin�k�� � �k cos�k���� �����

Such representation is unique� since the functions �� cos�� sin�� cos ��� sin ��� � � �form a basis in the space of all trigonometric polynomials� As the degree oftrigonometric polynomial the maximal d was considered� such that �dor �d�� is not equal to zero� As free term we consider ��

In this chapter we show that for small degrees of trigonometric polynomi�als p� q �up to �� center condition implies composition condition� All thesecomposition conditions will be directly written down� Part of computationswas done together with M� Kiermaier� working on this summer project undermy supervision�

��� Computational approach

We proved the following theorem

Theorem� For p� q up to degrees �� composition condition is not onlysu�cient� but necessary for the center�

To prove it we will compute subsequently coe�cients �k�x� of Poincare

��

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return map

y�x� y�� � y� ��Xk��

�k�x� �yk� �

and after substitution x � �� we will look for conditions on coe�cientsi� �i under which �k���� � �� The integration of trigonometric functionsworks very slow in �Mathematica�� so to increase the speed of computationsa function integ was introduced� which replaced the standard Integrateprocedure of Mathematica

�RecursionLimit������

integ�y� � z��x��integ�y�x�integ�z�x�

integ�c� y��x��c integ�y�x ��FreeQ�c�x�

integ�c��x��c x ��FreeQ�c�x�

integ�x��n� �x��x��n�����n��� ��FreeQ�n�x�� n �� ���

integ�Sin�a� x��x���Cos�a x�a ���a ��FreeQ�a�x�

integ�Cos�a� x��x�� Sin�a x�a ��FreeQ�a�x�

integ�Sin�x��x���Cos�x �� �

integ�Cos�x��x�� Sin�x �

Then we use TrigReduce to reduce trigonometric expressions to the formwhere integ can be applied�

psi�x��i�

TrigReduce�integ�TrigReduce���i����psi�x�i���p�x��i����psi�x�i���q�x�x�

After computing rst several equations �k���� � � we prove that theseconditions imply composition representability of P ���� Q���� and hence im�ply center�Let us x notations we work with� Let us notice� that integrating �����

w�r�t� � and taking into consideration

Z ��

sin t dt � ��

Z ��

cos t dt � � we

got that p���� q��� do not have free terms� i�e�������������p��� �

deg p���Xi��

��i�� cos���� �i sin����

q��� �

deg q���Xi��

���i�� cos���� ��i sin�����

��

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which corresponds to the following P � Q �������������P ��� �

deg p���Xi��

��i�� sin��� � �i cos���� �i�

Q��� �

deg q���Xi��

���i�� sin��� � ��i cos���� ��i�

Notice� that in contrast to the polynomial case� the degree of trigonomet�ric polynomial does not change after integration or di�erentiation�

��� Center conditions for p q of small de�

grees

����� Center conditions for the case when either p or

q is of the degree zero

If degree p is equal to zero� then p is a constant� and as free term of p is zero�p � �� Then the equation ����� becomes

y� � qa�x�y��

Lemma ���� The condition Q���� � � is necessary and su�cient for theexistence of a center�

Proof� This equation can be solved explicitly

y�

y�� q�x��

��

��y���

� �Q�x�����

�y��Q�x� � Const�

substituting x � � we get Const ��

�y����Now its obvious� that the center

condition y��� � y���� is equivalent to Q���� � �� �If the degree of q is equal to one� then similarly we obtain the center

condition is P ���� � ��

Corollary� If p � � �q � ��� any trigonometric polynomial q �p� respec�tively� without free term de�nes center� Formally speaking� there exists acomposition with polynomial q �p� respectively��

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����� Center conditions for the case deg p��� � �� deg q��� ���

Lemma ���� For deg p��� � deg q��� � � center conditions coincide withthe composition conditions� i�e� there exists center i P ��� � kQ����

Proof� We get �P ��� � � sin��� � � cos���� �Q��� � �� sin��� � �� cos���� ��

After computing we get ������ � ������ � ���� � � only if ��� � ���� therefore this conditionis necessary for center� Hence � � k��� � � k�� for some k �� �� henceP ��� � kQ���� �

����� Center conditions for the case deg p��� � �� deg q��� ���

Lemma ���� For deg p��� � �� deg q��� � � center conditions coincidewith the composition conditions� There are three center components for thiscase

� � �� �� � �� �� � � �����A��

corresponding to the composition with W ��� � cos�� ��

� � �� �� � �� �� � � �����B��

corresponding to the composition with W ��� � sin��

�� �����

� �� ����

�� � ���

��������C��

corresponding to the composition

Q��� �����

P ���� ���� � ����

��P ����

��

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Proof� We get �P ��� � � sin��� � � cos���� �Q��� � �� sin��� � �� cos���� �� � �� sin���� � �� cos����� ��

After computing we get ������ � ������ � ���������� � ������� � ����� � ���� � ���� � ������������ �

����������� � ������ � ����� � ������� � ������ � ����� �

������� � ������ � ������ � ������ � ������� � ������ � ��������

We assume that �� �� � � �� �� � ��� �� � �� �� ���

�� � � if either

Case �� �� �����

and � �� ��orCase �� � � � � In this case � � � � � is not a solution because we getdeg p � �� so we left with the only possibility �� � ��

Consider the rst case� Computing ������ � � under the written relation�we get ������ � ������ � � and ������ � ������� � ���

�� � �����

Solving the equation ������ � �� we get two cases

Case ��� �� �����

� �� ����

�� � ���

���and � �� � � � �� ��

Case ��� � � �� � �� � ��

Case �� Computing ������ � �� we get � � �� � �� � ��

�� � � for all the three cases� Hence we have obtained three sets of con�ditions on coe�cients� which are suspicious of being center conditions� Letsanalyze them�

Case ���In this case we get P ��� � � cos�� ��Q��� � �� cos� � �� � �� cos �� � �� � �� cos� � �� � ���� cos

� � � �� ����cos� � �� � �����cos� � ��� � ��cos� � ��� � ��� � �����cos� � �� �

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����cos�� ����So there is a composition with W ��� � cos� � �� �P �W � � �W and�Q�W � � ��� � ����W � ���W

�� so this condition is a composition con�dition and therefore this condition is su�cient for center�

Case �In this case we get P ��� � � sin��Q��� � �� sin� � �� cos �� � �� � �� sin� � ����� sin� � � �� � �� ��� sin�� ��� sin� ��So� in this case we get P and Q are compositions with W ��� � sin�� thiscondition is composition condition and hence it is su�cient for the existenceof a center�

Case ���In this case we getP ��� � � sin��� � � cos���� �

Q��� �����

sin��� � �� cos���� �� � �� sin���� ����

�� � ���

���cos����� ���

�� � ���

���

Lets look for the composition condition Q��� � aP ���� � b P ����

Substituting P and Q into this equation� we get �a������ sin��x�����

�a���

��a������ cos��x���b���a������ sin�x��

�b� � �a�� � ��� cos�x� � ���a�� � b� �

��a�� � �� � ��� � �

Solving this system� we obtain

a �����

� b ���� � ����

��

For this values of a and b we veried that aP ���� � b P ���� Q��� � ��and therefore in the case ��� center condition coincide with the compositioncondition� �

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����� Center conditions for the case deg p��� � �� deg q��� ��

Lemma ��� For deg p��� � �� deg q��� � � center conditions coincidewith the composition conditions� There are four center components for thiscase

� � �� � � �� �� � �� �����A�

corresponding to the composition with W ��� � cos�� ��

� � �� � � �� �� � �� �����B�

corresponding to the composition with W ��� � sin�� and

� ������

� � ����

�� � ����

������ �����C�

corresponding to the composition

P ��� ������

Q���� ���� � ����

����Q����

Exchanging k and �k� we get that these representation coincides with thecomposition representation for deg p��� � �� deg q��� � �� As the composi�tion condition is symmetric for these two cases� so the composition represen�tations must be symmetric as well�

Proof� We get �P ��� � � sin��� � � cos���� � � � sin���� � � cos����� �Q��� � �� sin��� � �� cos���� ��

After computing we get

����� � ������ � ��������� � ������� � ���� � ���� � ����� � ���� � ����������� �

��������������������������������������������������

������ � ����� � ����� � ������� � ������� � ������ � ������� �

��

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������� � ������� � ������ � ������ � ����� � ������

We assume that �� �� � � �� �� and ��� �� � �� �� ���

����� � � i� at least one of the three conditions is satised

Case �� � ������

� �� �� �

Case �� � � �� � �

Now we compute ������ � � for these cases

In the case � we get ������ ���

���������������������� so we obtain

the following cases

Case ��� � ����

�� � ����

������ � �

�����

� �� �� �� �� �� ��Here

P ��� ������

sin��� � � cos���� � � � sin���� ������

cos����� �����

Q��� � �� sin��� � �� cos���� ���solving the equation P �x� � aQ�x�� � bQ�x� � � w�r�to a� b� we found a ������

� b ���� � ����

��� and for these values of a� b P �x��aQ�x���bQ�x� �

�� So� this case is composition case�

Case ��� � � � � �� � � ThenP ��� � � cos����� cos ���� � ���cos�������������l���cos������� Q��� � ���cos� � ��� so we get composition with W ��� � cos��� � ��hence it is center condition�

In the case � we get

������ � ������� ������ ��

��������

�� � ���� � ��� �m��� and

hence from ������ � � follows � � � � � � �� � �� From the thirdsolution follows p��� � �� hence this case is impossible� and we are left withthe two cases

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Case ���� � � � � �� � �� and in this case also ������ � � and wegetP ��� � � sin� � � cos �� � � � ��� sin� � � � sin�� Q��� � �� sin��so P and Q are compositions with W ��� � sin��

Case ���� � � � � �� � �� and then from ������ � � follows � � ��so this case coincide with ���� �

����� Center conditions for the case deg p��� � �� deg q��� ���

In this case P and Q are of the form

P ��� � � sin��� � � cos���� � � � sin���� � � cos����� �

Q��� � �� sin��� � �� cos���� �� � �� sin���� � �� cos����� ��

Without loss of generality we can shift coe�cients p and q of the Abelequation on the angle � p�x�� � p�x� ���q�x�� � q�x� ���Then

Pnew��� �

Z �

pnew�x�dx �

Z �

p�x � ��dx � P �x� ��� P ����

and similarly Q�

Shift on the angle � P �x����P ��� � �� cos� sin��� cos� sin� cos �x�sin���sin� cos x��cos���cos� cos x��cos� ���sin� ���cos� � cos �x��sin� � cos �x��cos�� sinx�sin�� sinx�cos� � sin �x�sin� � sin �x�� cos� sin�� sin �x �cos x�� sin� � � cos�� � cos �x�� cos� sin� � � cos

� �� � sin� �� � � � ��

so we can kill any term by a shift on an appropriate angle ��Using rotation� we can put any coe�cient being equal to zero� let � � ��

then using rescaling� we get � � � �both of them can not be zeroes� sincethe degree of P is ���

��

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Lemma ���� For�P ��� � � sin��� � � cos���� � � cos����� �Q��� � �� sin��� � �� cos���� �� � �� sin���� � �� cos����� ��

the necessary and su�cient conditions for the center is either P is propor�tional to Q� or both of them are functions only of sin� or cos��

Proof� From ������ � ������ � ��� � ���� � �� we nd �� � ������ �

����� substituting it into ������ we get������ � ����

������������������

�����������

������������

Solving it� we have several options�

Case I� � � �� then ������ ��������

�� � ��� so either � � � �Case

I���� or �� � � �Case I���� or � � �p� �Case I����

Case II� � � �� then ������ ��������

��� ���� so either � � � �Case

II���� or �� � � �Case II���� or � � �p� �Case II���� Case II��� coincides

with the Case I���

Case III� � �� �� � �� �� then from ������ � � we get

�� � � �

������� � ������ � ���� � ���� � ����� � �

�����

Substituting it into the program� we get

������ � � �

��������� � ����

�� � ��������� � ��� � ������� � ��� �

��������������������������������������������������������� �

������ � ���

����� � �� � ��

��� � ������ � ��� � ���� � ��� � ��� � �����

���� � �� � ���

��

Solving the equation ������ � � with respect to ��� we get the followingcases

��

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Case III�� �� �����

� and this is the only solution if �� � �� � ��

Case III�� If �� � �� �� �� there is another solution

�� ������� � ��� � ���� � ��� � ��� � �����

���� � �� � ����

In the Case III�� we get

����� ������� � �� � ���

��� � �� � �������� ���� � ����� � ���� � ������� � ��� �

���� ��� � ����� � ���� � �������� � �������

We can nd �� from the equality ����� � �� it is possible only for

�� � ���� ���� � �

���� � ���� � ������� � ���

���� ��� � ��� � ���� � ������ � ����

For these values of � and all polynomials up to ����� vanish� and wecompute ������ � �������������After standard computation of resultants �see �B�� or Chapter � for shortexplanation of resultants technique�� we obtain that these three polynomialsdo not have common zeroes�

Lets consider the rest of all of these cases�In the Case I��� � � � � � � �� � � �� then �� � �� and�

P ��� � cos����� �Q��� � �� sin��� � �� cos���� �� � �� cos����� ��

computing ������ we get either �� � �� or �� � �� and in both cases we getthe composition � with cos� and sin� respectively�In the Case I��� � � �� � � � �� � � �� then �� � ���

P ��� � � cos���� � � cos����� �Q��� � �� cos���� �� � �� cos����� ��

and we get composition with cos��In the Case I��� � � � � �� � � �� � �

p�� �case � � �p� is

similar��P ��� �

p� cos�����p� � cos����� �

Q��� � �� sin��� � �� cos���� �� � �� sin���� � �� cos����� ���

��

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Performing computations� we get �� � �� and it is a composition with cos��In the Case II��� � � �� � � � �� � � �� then �� � ���

P ��� � � sin���� � cos����� �Q��� � �� sin��� � �� cos����� ��

and we get composition with sin��In the Case II��� � � � � �� � � �� � �

p�� �case � � �p� is

similar��P ��� �

p� sin���� � cos����� �

Q��� � �� sin��� � �� cos���� �� � �� sin���� � �� cos����� ���

Performing computations� we get �� � �� and it is a composition with sin��

In the Case III��� � � �� � � �� �� �����

� and then �� � �� so

�P ��� � � sin��� � � cos���� � � cos����� �Q��� � ����

��sin��� � �� cos���� �� � �� cos����� ��

Then from ������ � � we get either � � �� and then P � Q are compositionswith cos�� or �� � ���� then �� � ���� and Q � ��P � i�e� we again getcomposition� �

��

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Chapter �

Center and CompositionConditions for Abel di�erentialequations with p� q � algebraicpolynomials of small degrees

��� Introduction

In this chapter we summarize results on Abel equation with p� q � algebraicpolynomials� which were obtained in �BFY���� BY��� in the authors M�Sc�thesis and in the current Ph�D� thesis� All the results� for which an explicitreference is not given� are new and belong to the current thesis� so they aregiven with proofs�We consider Abel di�erential equation

y� � p�x�y� � q�x�y� �����

with p�x�� q�x� � algebraic polynomials in x� Although this problem does notcorrespond directly to the classical center problem on a plane �where p� q aretrigonometric polynomials�� it represents a signicant interest by itself� as ithelps to understand the combinatorial structure of the set of closed solutions�A center condition for this equation �closely related to the classical centercondition for polynomial vector elds on the plane� is that for xed points� and a and for any solution y�x� y� � y��� � y�a� � i�e� all the solutionsreturn to the initial value on the interval ��� a��

��

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As before� we are looking for solutions of ����� in the form

y�x� y�� � y� ��Xk��

vk�x� �yk� �

where y��� y�� � y�� � ��� �� � � �� is the �nite� set of the coe�cients of p�q� Shortly we will write vk�x��The coe�cients vk are computed by formula ����� and turn out to be

polynomials both in x and � We study the polynomials ideals Ik � R� � x��Ik � fv��x�� v��x�� � � � vk�x�g� The problem is to �nd conditions on p qunder which x � a is a common zero of all Ik�

Following �BFY�� we introduce periods of the equation ����� as those� � C � for which y��� � y��� for any solution y�x� of ������ The generalizedcenter conditions are conditions on p� q under which given a�� � � � � ak are�exactly all� the periods of ������The ideal I � C �� x� is studied� where is a set of coe�cients of poly�

nomials p� q�

I � fv��x�� v��x�� � � � vk�x�� � � �g ��k��

Ik� where Ik � fv��x�� v��x�� � � � vk�x�g�

Our generalized center problem is the following For a given set of dierent complex numbers a� � �� a�� � � � � a� �nd conditionson algebraic polynomials p� q� under which these numbers are common zeroesof I�We say that ����� denes center on the set of numbers a�� � � � � a� � Thesenumbers are called periods of ������ since y��� � y�ai� for all the solutionsy�x� of ������

��� Composition conjecture for multiple ze�

roes and its status for small degrees of p

and q

The following composition conjecture has been proposed in �BFY��

I ��k��

Ik has zeroes a�� a�� � � � ak a� � � if and only if

��

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P �x� �

Z x

p�t�dt � �P �W �x�� Q�x� �

Z x

q�t�dt � �Q�W �x��

whereW �x� �kYi��

�x� ai� �W �x� is a polynomial vanishing at a�� a�� � � � ak

and �P �Q are some polynomials without free terms �W �x�is an ar�bitrary polynomial��

Su�ciency of this conjecture can be shown easily �see �BFY���� But westill do not have any method to prove the necessity of this conjecture inthe general case� although the connection between this conjecture and someinteresting analytic problems was established �see �BFY��� �BFY��� �BFY����and for some simplied cases it was partially or completely proved�The following work was started during M�Sc� thesis �B�� and was com�

pleted during Ph�D� thesis

a� The maximal number of di�erent zeroes of I� i�e� the maximal numberof periods of ����� was estimated

Theorem ����� Either the number of surviving dierent zeroes �including�� of I is less or equal then �degP � degQ��� or P is proportional to Q�

Corollary� Either P is proportional to Q� or the number of dierent peri�ods of ����� is less than or equal to ��degP � degQ���� ��

This result was proved in M�Sc� thesis and is implicitly contained in com�putations� given in �BFY�� �

The next results were obtained in M�Sc� thesis �B�� BY���b� The above stated composition conjecture was veried for the following

cases �degP� degQ� � ��� ��� ��� ��� ��� ��� ��� ��� ��� ��� ��� ��� ��� ��� ��� ��� It wasdone using computer symbolic calculations with some convenient represen�tation of P and Q � Computations for the cases �degP� degQ� � ��� ��� ��� ��were performed together with Jonatan Gutman and Carla Scapinello� work�ing on this summer project under my supervision�

��

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Namely� the following theorem was proved

Theorem� For the following cases the composition conjecture is true andthe following table gives the possible number of dierent periods in each case

deg P � � � � �deg Q

� �� � � �� � � �� �� � �� � �� �� � � �� �� �� �� �� � �� �� �

The proof consists of computations of several rst �n�x� for each of thecases considered� equating �n�x� to zero and solving the resulting systemsof polynomial equations on coe�cients of p and q and on a� It was done us�ing computer symbolic calculations using the special representation of P andQ� In most of the cases straightforward computations were far beyond thelimitations of the computer used� Consequently� some non�obvious analyticsimplications were used�

For computations we used resultants technique� Resultants provide uswith a convenient tool for checking� whether n�� polynomials of n variablesPi�x�� � � � � xn� � C � x�� � � � � xn� do not have common zeroes�Consider one example� Assume we are interested whether polynomials

P �x� y�� Q�x� y�� R�x� y� have common zeroes�

Claim� Let Resultant�P�Q� x� � S��y�� Resultant�R�Q� x� � S��y�� IfResultant�S�� S�� y� �� �� then P � Q� R do not have common zeroes�

Proof� Assume there exists common zero �x�� y�� of all polynomialsP�Q�R� then S��y�� � S��y�� � �� hence Resultant�S�� S�� y� � �� Contradiction��

The general construction for n � � polynomials of n variables is exactlythe same�

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c� On this base explicit center conditions for the equation ����� on �����were written in all the cases considered� They turned out to be very simpleand transparent� especially in comparison with the equations provided byvanishing of vk��� �� See section ���� for detailed description of the centerset�

��� Description of a center set for p q of

small degrees�

Consider again the polynomial Abel equation

y� � p�x�y� � q�x�y�� y��� � y�

with p�x�� q�x� � polynomials in x of the degrees d�� d� respectively� We willwrite

p�x� � d�xd� � � � �� ��

q�x� � �d�xd� � � � �� ���

�d� � � � � � �� �d� � � � � � ��� � �� �� � Cd��d����

Remind that vk�x� are polynomials in x with the coe�cients polynomiallydepending on the parameters �� �� � C d��d���� Let the center set C C d��d��� consist of those �� �� for which y��� � y��� for all the solutionsy�x� of ������Clearly� C is dened by an innite number of polynomial equations in

�� �� v���� � �� � � � � vk��� � �� � � � In other words� C is the set of zeroesY �I� of the ideal I � fv����� � � � � vk���� � � �g in the ring of polynomials C �� ���In this section we consider I as the ideal in C �� �� and not in C � x� � �� �We x one endpoint a� say a � ��The table of multiplicities� obtained in �B�� BY��� gives the number of

equation vk��� � �� necessary to dene C �i�e� the stabilization momentfor the set of zeroes of the ideals Ik�x��� Since both vk��� and �k��� arepolynomials of degree k � � in �� ��� the straightforward description of Ccontains polynomials of a rather high degree� for example up to degree �� of� variables for the case �degP� degQ� � ��� ���

��

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The composition conjecture� in contrast� gives us very explicit and trans�parent equations� describing this center set C� Especially explicit are equa�tions in a parametric form �see below��

����� The central set for the equation ����� with deg p � deg q � � hasbeen described in �BFY��� We remind this result here� Let

p�x� � �x� � �x � �

q�x� � ��x� � ��x � ��

Theorem ����� ��BFY� � Theorem V��� The center set C � C � of the Abelequation ����� is given by

�� � �� � �� � ���� � ��� � ��� � �

��� � ��� � �

The set C in C � is determined by vanishing of the �rst � Taylor coe�cientsv���� � �� � � � � v���� � �

���� Now let

p�x� � �x� � �x

� � �x� �q�x� � ��x � ��

Theorem ����� The center set C � C � of the Abel equation ����� is givenby

�� � �� � ��� � � � �� � �

�� � ��� � �

The set C in C � is determined by vanishing of the �rst � Taylor coe�cientsv���� � �� � � � � v���� � ��

Proof� By the composition conjecture� which holds for this case� p andq belong to the center set if and only if P � �P �W �� Q � �W � whereW � x�x � ��� So� we may assume Q � �x�x � ��� P � �W � � W � Thuswe get

Q � �x�x� �� � ���x� � ��x

��

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P � � �x�x� ���� � x�x� �� � ��x� �

��x� �

��x� � �x

Comparing coe�cients of xk in both sides of equalities� we get���

� � �� � � ���� � �� � � � � � �� � �� �� � ��

which is equivalent to the system in the statement of the theorem��

����� Ifp�x� � �x � �q�x� � ��x

� � ��x� � ��x� ��

then similarly to the previous theorem one can prove the following

Theorem ����� The center set C � C� of the equation ����� is given by

��� � ��� � ���� � �� � ��� � �

� � �� � �

The set C in C � is determined by vanishing of the �rst � Taylor coe�cientsv���� � �� � � � � v���� � ��

���� Now let

p�x� � �x� � �x

� � �x� � �x

� � �x � �q�x� � ��x � ��

Theorem ���� The center set C � C of Abel equation ����� is given by

�� � �� � ���� � ��� � ��� � ��� � ��� � ��� � �

� � �� � ��� � ��� � ��� � ��� � �

The set C in C is determined by vanishing of the �rst � Taylor coe��cients v���� � �� � � � � v ��� � ��

��

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Proof� By the composition conjecture� which holds for this case� p andq belong to the center set if and only if P � �P �W �� Q � �W � whereW � x�x� ��� So� we may assume Q � �x�x� ��� P � �W � � W � � �W �Thus we get

�x�x� �� � ���x� � ��x

� �x�x� ����� �x�x� ������x�x��� � ��x��

��x��

��x��

��x��

��x���x

Hence �������

� � �� � � ����� � ��� � � � � ���� � � � � � �� � � ���� � �� �� � ��

which is equivalent to the system in the statement of the theorem��

����� If

p�x� � �x� �q�x� � ��x

� � ��x� � ��x

� � ��x� � ��x� ��

then similarly to the previous theorem one can prove the following

Theorem ����� The center set C � C of the Abel equation ����� is givenby

��� � ��� � ����� � ���� � ���� � ���� � ���� � ���� � �

�� � ��� � ���� � ���� � �� � �� � �

The set C in C is determined by vanishing of the �rst � Taylor coe�cientsv���� � �� � � � � v���� � ��

����� Now let

p�x� � �x� � �x

� � �x� ��

q�x� � ��x� � ��x

� � ��x � ���

��

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Theorem ����� ��BFY� � Theorem ����� The central set C � C of Abelequation ����� consists of two components C��� and C���� each of dimension��

C��� is given by ��� � �� � �� � ��� � �

��� � ��� � ��� � ���� � ����������

and ��� � ��� � ���� � ��� � ���� � ��� � �

���������

and C��� is given by ��������� and��� � �� � ���� � ��� � �

���������

The set C in C is determined by the vanishing of the �rst � Taylorcoe�cients v���� � �� � � � � v ��� � ��

This theorem was proved in �BFY�� using the fact that the compositionconjecture is true for this case� The component �������� �������� corre�sponds to the proportionality of P and Q� and the component �������� �������� corresponds to the composition with W � x�x� ���

����� Let

p�x� � �x� � �x� �

q�x� � ��x� � ��x

� � ��x� � ��x

� � ��x� ��

then similarly to the previous theorems one can prove the following

Theorem ����� The center set C � C of the Abel equation ����� is givenin a parametric form by

�����������

� � � � � ���a � ��� � a �� � ���� � �����a� �� �� � ���a� ��

� � �a��� � ���a�a � �� � � �� � ��a

� � ��a� �� �� � a

���������

��

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or by ���������������������������������

�� � �����

����

� �

�� �����

����

� �

��

� �����

�� � ������

����

� �

������

�� ����

��

��� �

����

� �

������

�� � ��� � ��

���������

The set C in C is determined by the vanishing of the �rst � Taylor coe��cients v���� � �� � � � � v����� � ��

Proof� We can represent P � W � Q � �W � � W � where W � x�x ����x� a�� Thus

�x� � �a� ��x� � ax

����x� �

��x� � �x�

��x� � �a � ���x� � a�x� � ��a� ��x� � �ax� � �a�a � ��x���

�x� � x��a� �� � ax

�����x� � � � �� ��x

Comparing coe�cients by xk in both sides of equalities� we obtain ����������After some transformations we obtain ���������� �Notice� that � �� � asleading coe�cient�� �

��� Composition conjecture for some special

families of polynomials

In this section we generalize M�Sc� thesis results� nding out some classes ofpolynomials p� q of arbitrary high degree� for which condition for a center ona given set of numbers can be explicitly found �and coincide with compositioncase��

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Let a� � �� a�� � � � � a� be given� Consider any polynomialW �x� vanishingat all the points aj� j � �� � � � � ��

Theorem ����� Assume that for at least one aj�

Z aj

W kdx �� � andZ aj

W ndx �� �� Polynomials p � W k�� � W ��� q � W n�� � �W �� de�ne

center on ��� a�� � � � � a�� if and only if � � � � ��

Remark� Notice� that the condition �

Z aj

W kdx �� � for at least one aj�

is satised� for instance� for W �x� �

�Yi��

�x� ai�� where all aj are di�erent�

Indeed� consider the function f�x� �

Z x

W �t�kdt� If all aj� j � �� � � � � � would

be zeroes of f�x�� then deg f � �k����� But degW � �� so deg f�x� � k����We obtain k� � � � �k � ���� which is not satised for � � � �Similarly one can show that W �x� �

�Yi��

�x� ai�mi satises the condition

Z aj

W kdx �� � for at least one aj� for almost all k� and so on� So� this

condition is �almost generic��

Proof� Since ���x� � P �x�� the conditions ���aj� � � imply � � ��Since ���x� � P ��x��Q�x�� the conditions ���aj� � � imply � � �� �

Theorem ����� Assume that degW � � and for at least one aj

det

�������

Z aj

W ndx

Z aj

W nW ��dxZ aj

W n�k��dx

Z aj

W n�k��W ��dx

������� �� ��

Polynomials p � W k�� � W ��� q � W n�� � �W � � �W ��� de�ne center on��� a�� � � � � a�� if and only if � � � � � � ��

Proof� The conditions ���aj� � � imply � � �� The conditions ���aj� �

��

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� imply

Z aj

W n � �

Z aj

W nW �� � ��

and the conditions ���aj� � � imply

Z aj

W n�k�� � �

Z aj

W n�k��W �� � ��

If the determinant of the system is nonzero� we get that the system has theonly zero solution� �

��

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Chapter

Riemann Surface approach tothe Center problem for Abelequation

��� Introduction

In this chapter Riemann Surface approach to the Center problem for Abelequation is discussed� This is a convenient general setting� where Abel Equa�tion

dy

dz� y�

dP �z�

dz� y�

dQ�z�

dz

considered on a given Riemann Surface� In this chapter

� we generalize notions of center and composition to the case of Riemannsurfaces�

� following Chapter �� we show that all the facts about Poincare returnmap and recurrence relations to compute its coe�cients remain validin this setting�

��� Abel equation on Riemann Surfaces

Let X be a domain on a connected Riemann Surface and let P and Q be twoanalytic functions on X� For Abel equation related to planar vector elds X

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is a neighborhood of the unit circle S� on C � Laurent polynomials P and Qare analytic on X� We consider the following Abel di�erential equationon X

dy � y� dP � y� dQ �A�

A �local� solution of �A� is an analytic function y on an open set ! in X�such that the di�erential forms dy and y� dP � y� dQ coincide in !�If x is a local coordinate in !� �A� takes the usual form

dy

dx� y� p�x� � y� q�x��

where

p�x� �d

dxP �x�� Q�x� �

d

dxQ�x��

Let Y � X be the universal covering ofX� The equation �A� can be liftedonto Y � One can easily show� that for any a � Y and for any c � C � thereis a unique solution yc of �A� on Y � satisfying yc�a� � c� whose singularitiestend to innity as c tends to zero� In what follows we always assume thatc is su�ciently small� so yc is regular and univalued on any compact part ofY � but in general is multivalued on X�

De�nition ����� Let � be a closed curve in X� We say that the Abel equa�tion �A� has a center along the curve �� if for any small c � C yc isunivalued along ��

Notice that in this denition it is su�cient to assume that for any su��ciently small c and some a � �� yc�a� � yc���a�� where yc�� is a result of ananalytic continuation of yc along �� Indeed� by uniqueness of a solution of therst order di�erential equation �A�� yc�a� � yc���a� implies that yc and yc��coincide in a neighborhood of a� Then by analytic continuation yc coincidewith yc�� along the the whole �� i�e� yc is univalued�

De�nition ����� We say that �A� has a total center on X� if it has acenter along any closed curve in X�

In particular� this is always the case for X � simply�connected� Indeed�in this case any closed curve is homotopic to a point� so after analytic con�tinuation along any closed curve we will obtain the initial value at this point�

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De�nition ����� Let �X be a domain on another Riemann Surface� �P and�Q be analytic functions on �X� Assume there is an analytic mapping w X � �X� such that P �x� � �P �w�x��� Q�x� � �Q�w�x��� We say that the Abelequation �A� on X is induced from the Abel equation

dy � y� d �P � y� d �Q � �A�

on �X by the mapping w� We also say that �A� is factorized through �X�

Notice� that the words �factorized through w X � �X� are equivalent tothe words �composition representation P �x� � �P �w�x��� Q�x� � �Q�w�x�����When we deal with Abel di�erential equation� we shall use the word �factor�ization�� and when we deal with P and Q� we shall use the word �composi�tion��

Lemma ����� Let �A� be induced from � �A� by w� Then any solution y of�A� is induced from a corresponding solution �y of � �A�� i�e� y � �y � w�

Proof� Let y � yc take a value c at some point a � X� Consider the solution�yc of � �A�� taking the value c at w�a� � �X� By the �invariance of the rstdi�erential�� �yc � w is a solution of �A�� and it satises �yc � w�a� � c� Hencelocally� �yc � w � yc� and analytic continuation completes the proof� �

Corollary ����� If �A� is induced from � �A� by w� and if � �A� has a centeralong w���� then �A� has a center along ��

De�nition ���� Let a and b be two dierent points in X� We say thata and b are conjugate with respect to the equation A� and withrespect to a certain homotopy class of curves � joining a and bin X� if for any solution yc of �A� with su�ciently small c �yc�a� � c�� itscontinuation yc�� along � satis�es yc���b� � c� In other words� any solutionof �A� takes equal values at a and b after analytic continuation along ��

A priori it is not evident that these denitions are natural and that con�jugate points can appear at all� However� the following proposition gives abasic reason for their appearance

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Proposition ����� Let X� �X� P � Q� �P � �Q� w be as above� Consider twodierent points a� b � X and a path � joining them� If w�a� � w�b� and � �A�has a center along the closed curve w���� then a and b are conjugate along� for the Abel equation �A�� In particular� if �X is simply connected� any twopoints a� b with w�a� � w�b� are conjugate along any � joining them�

Proof� Follows immediately from the last lemma� �

����� Example Polynomial Composition Conjecture

in the case X � C � P and Q polynomials in x

Here X is simply�connected� P and Q are analytic on X and hence the Abelequation

dy

dx� p�x� y� � q�x� y�� �����

with p�x� �d

dxP �x�� q�x� �

d

dxQ�x�� has a center along any closed curve ��

As far as a factorization of ����� is concerned� assume that there existpolynomials w� �P � �Q� such that P �x� � �P �w�x��� Q�x� � �Q�w�x��� Thenfor �X � C ����� is induced by w from

dy

dw� �P ��w� y� � �Q��w� y�� �����

By proposition ������ any two points a and b such that w�a� � w�b� areconjugate along any path ��In �BFY���� and �BY� this example was investigated in some details� In

particular� for small degrees of P and Q it was shown� that conjugate pointscan appear only in this way�

The following conjecture was proposed in �BFY��

Polynomial Composition Conjecture� Two dierent points a and b inC are conjugate for the Abel equation �A� with P � Q � polynomials in x� ifand only if the following Polynomial Composition Condition is satis�edthere exists a factorization P �x� � �P �w�x��� Q�x� � �Q�w�x�� with polyno�mial mapping w� such that w�a� � w�b��

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Notice� that although the equation �A� is non�symmetric with respect top and q� this conjecture proposes the symmetric condition if a and b areconjugate for �A�� then they are conjugate also for the equation

dy

dx� q�x� y� � p�x� y�� �����

This situation is not unique in the center problem� In �Che�� it was shownthat the Lienard system

d�y

dx�� f�x�

dy

dx� g�x� � �

has a center at the origin if and only if the functions F �x� �R x�f�t�dt�

G�x� �R x�g�t�dt can be represented as a composition F �x� � �F �z�x���

G�x� � �G�z�x�� for an analytic function z�x�� with z���� � ��

����� Example Laurent Composition Condition in the

case X a neighborhood of the unit circle S� �fjxj � �g � C � P and Q Laurent series� conver�

gent on X

We may rewrite Abel di�erential equation in an invariant form

d� � dP ��� �� � dQ��� ���

then expressing sin � and cos � through x � ei�� i�e����cos � � x�x��

��

sin � � x�x��

�i�

� � y�

we obtain P and Q in the form of Laurent polynomials in x� and thedi�erential equation is

dy � y� dP � y� dQ

considered on the circle x���� ���� � S��We shall discuss this case in much more detail in the next chapter� be�

cause it corresponds directly to the classical Center�Focus Problem for ho�mogeneous polynomial vector elds on the plane�

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The one of possible factorizations in this case� which we shall callLaurentComposition Condition� takes the form P � �P �w�� Q � �Q�w�� where wis a Laurent series� and �P � �Q are regular analytic functions on the disk D inC � containing the image w�S���

Lemma ����� Laurent Composition Condition implies center along S��

Proof� We have a factorization w X � D� and since D is simply con�nected� the Abel equation dy � y�d �P � y�d �Q on D has a total center� thenby Corollary ����� it implies center along S�� �

In this case the conjecture that this is the only reason for center is nottrue there are known cases of center along S� for the equation �A�� whenp and q can not be represented as a Laurent composition �the example wasconsidered by Alwash in �A� � see Chapter � above��Nevertheless� for p�z�� q�z� � Laurent polynomials of small degrees up to

������ i�e� of the form z���a�z� � a�z

� � a�z� � a�z � a��� the composition

representation is the only possible reason for center �shown in Chapter ���

��� Poincare mapping

Here we discuss notions introduced in Chapter �� in a more general setting�Let us return to a general situation of the Abel equation �A�

dy � y� dP � y� dQ

on a Riemann Surface X� Let a� b � X and let � be a curve in X� joining aand b�

Let us consider a general solution yc�� of the equation �A� along �� takingthe value c at the point a yc���a� � c� For su�ciently small c we mayrepresent it in a form of a �convergent� power series in c

yc���x� � c��Xk��

vk ck� �����

where vk � vk�a� x� �� are �multivalued� functions on X� x is a point on �

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For xed a� � substituting the solution yc���x� in the form of expansionyc���x� � c �

P�k�� vk�x� c

k into �A�� we obtain the following recurrencerelation on vk�x�

���������������

v��x� � �v��x� � �and for n � �vn�a� � � and

dvn�x� � dP �x�Xi�j�n

vi�x�vj�x� � dQ�x�X

i�j�k�n

vi�x�vj�x�vk�x�

�����

De�nition ����� The Poincare mapping "� �C � �� � �C � �� of the equa�tion �A� along � is de�ned by

"��c� � yc���b�� �����

where yc�� is� as above� the solution of �A�� taking the value c at the point a�and analytically continued along � up to the point b�

As it was shown�

"��c� � c��Xk��

vk�b� ck �����

For X � C � a � �� b � �� we obtain the usual Poincare mapping as weused in �BY��

Lemma ����� �� The equation �A� has a center along unclosed curve � withend points a and b �i�e� points a and b are conjugate� if and only if vk�b� � �for any k � �� Here vk�x� �for �xed a� �� are obtained by integration along� of �������� The equation �A� has a center along closed curve � if and only if vk�x�are univalued functions along ��

Proof� Follows immediately from denitions of a center and Poincare map�ping� �

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The recurrence relations ����� can be in fact linearized in the followingsense Let us x the curve �and the end�points a and b� and consider aninverse function �with respect to y and c� to yc���x�

c � y���x� yc���x�� � y���x� y�

One can easily see that for su�ciently small c and y we can represent y�� bya convergent power series

c � y���x� y� � y ��Xk��

�k�x�yk� �����

where �k�a� � �� since y���a� c� � c�

Lemma ����� BFY�� The coe�cients �k�x� satisfy the following recur�rence relation��

����x� � �� ���x� � � and for k � �d�k�x� � ��k � ��dP �x��k���x�� �k � ��dQ�x��k���x��k�a� � �

�����

Proof� Write ����� as

c ��Xk��

�k�x�yk� �� � �� �� � � �����

Taking a di�erential of it with respect to x along the solution of �A�� we get

� ��Xk��

d�k�x�yk �

�Xk��

k�k�x�yk�� � dy �

�Xk��

d�kyk�

�Xk��

��k � ��dP �x��k���x� � �k � ��dQ�x��k���x��yk

Equating to zero the right�hand side coe�cients produces the required re�currence� �

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De�nition ����� The Inverse Poincare mapping "��� �C � �� � �C � ��

of the equation �A� along � is de�ned by

"��� �c� � y�c���b�� �����

where y�c�� is� as above� the solution of �A�� taking the value �c at the point a�and �c is chosen such that yc���b� � �c� Shortly� " �"�� � id�

As it was shown�

"����c� � �c��Xk��

�k�b� �c �����

Notice that in our considerations we were free to choose the endpoint b�It leads immediately to the following

Lemma ����� BFY�� For any q � �� for any b � C the ideals Iq �

fv��b�� � � � � vq�b�g and #Iq � f���b�� � � � � �k�b�g coincide�

Proof� This follows immediately from the standard expressions for theTaylor coe�cients of the inverse function if y � x � a�x

� � a�x� � � � ��

x � y � b�y� � b�y

� � � � �� then b� � �a�� b� � �a� � �a��� � � �� a� � �b��a� � �b� � �b��� � � � �

Now we get as the corollary the following theorems

Theorem ����� The equation �A� has a center along an unclosed curve �with end points a and b �i�e� points a and b are conjugate� if and onlyif �k�b� � � for any k � �� Here �k�x� �for �xed a� �� are obtained byintegration along � of the linear recurrence ������

Theorem ����� The equation �A� has a center along a closed curve � if andonly if all the functions �k�x� are univalued functions along ��

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Chapter

Factorization �composition� forthe case X � C � P and Q �rational functions

�� Introduction

We started from the center problem for vector elds on the plane� It wasreduced to the center problem for Abel di�erential equation with p� q �Laurent polynomials� The natural question is to study Abel equation withp� q � rational functions� Everything� what can be said in general aboutrational functions� remains valid for the case of Laurent polynomials�It turns out that the assumption that P and Q are rational functions

puts the composition �factorization� question completely in the frameworkof algebraic geometry of rational curves and of the Ritt theory ��R���� Weshow that the examples of Polynomial composition and Laurent composition�given in sections ����� and ������ are in some sense �generic�� and any analyticfactorization of rational function can be reduced to composition of rationalfunctions�

�� Composition and rational curves

The following facts are very basic in algebraic geometry� We restate themfor convenience of our presentation� For details we address the reader to any

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classical algebraic geometry text �e�g� �Sha���

Well denote by Y the curve in C � � parameterized by P � Q Y � f�P �t�� Q�t��� t � C g�

Lemma ����� The curve Y for rational P and Q is an algebraic curve�

Proof� We need to prove that for su�ciently large d there exists a poly�nomial F �x� y� of the degree d� such that F �P �t�� Q�t�� � �� Without lossof generality we may assume that P and Q have the same denominatorR� Consider all the products P �t�iQ�t�jRd� i � j � d� These are polyno�mials of t of the degrees less than or equal to i degP � j degQ � d degR �d�degP�degQ�degR�� But there exist d�d���

�such products P iQj� therefore

for d�d����

� d�degP�degQ�degR�� i�e� for d � ��degP�degQ�degR���there exists a linear dependence over C X

i�j

ijP �t�iQ�t�jR�t�d � �� hence

Xi�j

ijP �t�iQ�t�j � ��

This polynomial F �x� y� �P

ijxiyj vanishes on Y � Such polynomial of

minimal degree denes an algebraic curve Y � �

L�uroth theorem� Any sub�eld of a �eld of rational functions is generatedby a rational function�

Corollary ����� There exist rational functions r�t�� �P � �Q� s�t� P �t� ��P �r�t��� Q�t� � �Q�r�t�� and s�t� the map

�� C � Y� z �� � �P �z�� �Q�z��

de�nes a birational isomorphism between C and Y � In particular� Y is arational curve�

Proof��� Notice that K � C �P �t�� Q�t�� is is a subeld of the eld of rationalfunctions C �t�� By L$uroth theorem K � C �r�t�� for some rational functionr�t�� In particular� P and Q belong to K� hence there exist rational functions�P �t�� �Q�t� such that P �t� � �P �r�t��� Q�t� � �Q�r�t��

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�� It is obvious that �� is surjective and rational� Lets prove that there existsan inverse map ���� Y � C � Since r�t� � C �P �t�� Q�t��� there exist a ra�tional function R�x� y� s�t� R�P �t�� Q�t�� � r�t�� i�e� R� �P �r�t��� �Q�r�t��� �r�t�� Obviously it is a rational function from Y to C � Let us prove thatR � ����� Indeed� R � �� � id C � C R����z�� � R� �P �z�� �Q�z�� �R�P �r�t��� Q�r�t��� � r�t� � z� since for any z there exists t z � r�t��Vice versa �� � R � id Y � Y � since ���R�P �t�� Q�t��� � ���r�t�� �� �P �r�t��� �Q�r�t��� � �P �t�� Q�t��� �

De�nition ����� The degree of a map � � �P�Q� C � Y is the degreeof the algebraic extension �C �t� C �P�Q���

De�nition ����� The parameterization of a rational curve Y � � C � Yz �� �P �z�� Q�z�� is called minimal if deg � � ��

From corollary ����� it follows that a minimal parameterization denes abirational isomorphism between C and Y �

De�nition ����� The mapping f X � Y is called �not ����� if thereexists an open set ! � Y � s�t� each point of ! has more than one preimageunder f �

De�nition ���� A rational function r is called common divisor undercomposition of rational functions P and Q� if P � �P �r�� Q � �Q�r�� Thecommon divisor r is called nontrivial� if r is not ����

De�nition ����� A rational function r is called Composition GreatestCommon Divisor CGCD� of rational functions P�Q� if r is a commondivisor under composition of P and Q� and if �r is another common divisorof P and Q under composition� then r � R��r� for a rational function R�

De�nition ����� The degree of a rational function is a maximum ofdegrees of numerator and denominator�

Lets notice that among rational functions only linear functions �functionsof degree �� are ���� Respectively� if we are looking for nontrivial CGCD inthe class of rational functions� it must have degree greater than �� It is easyto show that CGCD exists and satises all the properties of usual greatestcommon divisor� In particular�

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Proposition ����� For any rational functions P �t�� Q�t� their CGCD r�t�exists and is given by corollary ������ CGCD is unique in the algebra ofrational functions under compositions up to composition with an invertiblerational function �i�e� a function of degree ��� i�e� two CGCD of a givenfunction can be obtained each one from another by a �right or left� composi�tion with a linear function�

Proof�Obviously� r�t� from Corollary ����� denes rational common divisor undercomposition� If �r is another composition common divisor of P and Q� thenP � �P ��r�� Q � �Q��r�� so C ��r�t�� � C �P �t�� Q�t�� � C �r�t��� hence r�t� �R��r�t�� for some rational function R� Therefor r�t� is actually a CGCD�If r and �r are two CGCD� then r�t� � R��r�t��� but �r�t� � �R�r�t��� so

R � �R � id� hence R is a linear function� �

The following facts are proved� for example� in �Sha�

Lemma ����� �� For a rational map � � �P�Q� C � Y the number ofpreimages of almost each point is equal to deg ���� �C �t� C �r�t��� � deg r�t��

Corollary ����� The degree of the map � � �P�Q� is equal to the degreeof the rational CGCD of P and Q� If r is CGCD of P and Q� P � �P �r��Q � �Q�r�� then �� C � Y z �� � �P �z�� �Q�z�� is a minimal parameterization�

The following two results show that allowing an analytic �and not a priorirational� composition does not add� in fact� anything new�

Lemma ����� deg � � � if and only if there exists an analytic factorizationw C � �X� where �X is a Riemann Surface� �P and �Q are analytic functionson �X� such that P �t� � �P �w�t��� Q�t� � �Q�w�t��� and w is not ����

Proof�Let deg � � �� then taking w � r C � C we get the required analytic fac�torization� Vice versa� let there exist an analytic factorization w C � �X�P �t� � �P �w�t��� Q�t� � �Q�w�t��� Then C �P�Q� � C � �P �w�� �Q�w��� whichis a proper subset in C �t�� because w�t� glues some points in C � but inC �t� there are functions which map these points into di�erent points� Hence�C �t� C �P�Q�� � �� �

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Corollary ����� If there exists an analytic factorization of rational func�tions P � �P �w�� Q � �Q�w� with w � not ���� then there exists a nontrivialCGCD of P and Q P � �P �r�� Q � �Q�r� for �r of the degree greater than ��

�� Structure of composition in the case X �

C P and Q � polynomials

We shall prove that the factorization in the form of Polynomial Compo�sition Condition is essentially the only one natural factorization in thepolynomial case� namely

Theorem ����� Assume there exists an analytic factorization P � �P �w��Q � �Q�w� with w � not ���� Then there exists a polynomial factorizationP �t� � #P � #w�t��� Q�t� � #Q� #w�t��� with #P � #Q� #w being polynomials� the degreeof #w is greater than � and deg� #P� #Q� � ��

The proof will follow from the lemma

Lemma ����� If we have a factorization P � �P � r� Q � �Q � r with �P �t���Q�t�� r�t� � rational functions C � C � then there exists a linear rationalfunction C � C � s�t� �P � � �Q � � �� � r are polynomials�

Proof�This proof is contained essentially in �R�� Let r�� � a� Let the degrees ofthe rational functions r� �P be n� m respectively� The degree of the polynomialP will be mn� Then r�� � a with multiplicity not more than n� �P �a� � with multiplicitynot more than m� but �P � r�� � with multiplicity exactly mn� because�P � r is a polynomial�Hence we get that r�� � a with multiplicity n� �P �a� � with multi�

plicity m�Now take �z� � �

z�a� i�e� ���a� �� Then �P � �� � with multi�

plicity n� �� � r�� � with multiplicity m� hence they are polynomials�Similarly �Q � is a polynomial� �

Proof of theorem�By corollary ����� there exists a factorization P �t� � �P �r�t�� Q�t� � �Q�r�t��

��

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for some rational functions �P �t�� �Q�t� and rational function r�t� of degreegreater than �� such that deg� �P� �Q� � �� Then taking #P � �P �� #Q � �Q ��#w � �� � r we obtain the required polynomial factorization P �t� � #P � #w�t���Q�t� � #Q� #w�t�� with #w of degree greater than � and deg� #P � #Q� � �� �

The similar fact was proved by C� Christopher in �Chr�� when he investi�gated polynomial case of Lienard system�

Corollary ����� If deg�P�Q� � s � �� then the two polynomials P andQ have a nontrivial CGCD of the degree s in the algebra of polynomialsunder composition P �t� � #P �r�t��� Q�t� � #Q�r�t��� with #P � #Q� r � algebraicpolynomials and deg r � s� The map � z �� � #P �z�� #Q�z�� de�nes a minimalpolynomial parameterization of the algebraic curve Y � f�P �t�� Q�t��� t � C g�

�� Structure of composition in the case X �

a neighborhood of the unit circle on C P

and Q � Laurent polynomials

We describe possible factorization of Laurent polynomials� up to a naturalequivalence�

De�nition ���� The composition representations P � �P ��r� and P � �P ��r�are equivalent if there exists a linear rational function � s�t� �P � �P ����r � ����r��

Theorem ���� Up to the equivalence relation of de�nition ��� there areonly two types of composition representations of a Laurent polynomial P ��� P � �P �r�� where �P is a usual �algebraic� polynomial� and r is a Laurentpolynomial���� P � �P �r�� where �P is a Laurent polynomial� and r � zk for some k � N �k � ��Any two composition representations of types ��� and ��� for deg r � � andk � � are not equivalent�

Proof�

��

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P is a Laurent polynomial� hence has exactly two preimages � � and� Assume we are given a composition in the class of rational functions P � �P ��r� � We shall show that by choosing a suitable linear rational function we obtain P � � �P � � � ��� � �r�� where �P � �P � and r � �� � �r are ofthe required form ��� or ����

�P may have either two or one preimage of ��� Assume rst that has two preimages a �� b under the map �P

�P �a� � �P �b� � � Take a linear function �z�� s�t� ��� � a� �� � b

�z� ��

z � �a�b

� b� Then � �P � ���� �� � �P � ��� �� so �P � �P ��

is a Laurent polynomial� Then necessary ��� � �r���� � �� ��� � �r��� ��and there are no other points where ����r� takes values � and � so r �����r� is an algebraic polynomial of the form zk for some natural k��� If �P has only one preimage of � then similarly to the lemma ���

there exists a linear rational function s�t� �P � �P �� is a polynomial� Thennecessary �����r���� �� �����r��� �� and there are no other pointswhere ����r� takes values � and � so �� � �r is a Laurent polynomial�Since under composition with a linear function the number of preimages

of a given point can not change� ��� and ��� are not equivalent� �

Corollary ���� If deg�P�Q� � �� then P � �P �r�� Q � �Q�r� with either��� Laurent polynomial composition �P � �Q � algebraic polynomials� r �Laurent polynomial of degree greater than �� or��� �P � �Q � Laurent polynomials� r � zk for k � ��

��

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Chapter

Moments of P � Q on S� andCenter Conditions for Abelequation with rational p� q

��� Introduction

The role of generalized moments of the formRP iQjdP and

RP iq dz� as

related to the center�Focus problem and composition conditions on the in�terval� is investigated in �BFY����� �BY��� Recently a serious progress havebeen done in this direction ��Pa����� �N�� �Chr����In this thesis we restrict ourselves to the case of Laurent polynomials�

and generalized moments on the circle�It is shown that certain integral condition on P �

Rp and Q �

Rq

�namely vanishing of generalized momentsRjzj��

P iQjdP � �� impliescenter�

��� Su cient center condition for Abel equa�

tion with analytic p q

We return to the case of Abel equation �A�

dy � y�dP � y�dQ �A�

��

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considered in a neighborhood of a unit circle S� � fjxj � �g in the complexplane C � with P � Q � analytic functions in some neighborhood of S� �notnecessary Laurent polynomials��The following theorem is a summary of results due to J� Wermer ��W���

�W���� The applicability of Wermers results to Center problem was discov�ered by J��P�Fran%coise ��F��

Theorem Wermer ����� Let P � Q be a pair of functions on the unitcircle S� � C � Assume��� P and Q are analytic in an annulus containing S� and together separatepoints on S����� P � �� � on S����� P takes only �nitely many values more than once on S��

IfRs�P iQjdP � � for all i� j � �� then there exists a Riemann Surface

X and a homeomorphism � S� � X� such that ��S�� is a simple closedcurve on X bounding a compact region D� such that functions �P � �Q de�nedon ��S�� by P � �P � �� Q � �Q � � can be extended inside D to be analyticthere and continuous in D � ��S���

To use this theorem for our factorization� we need to replace �homeomor�phism� by �analytic map of a certain neighborhood of S� into X��

Lemma ����� Let S� � C be a unit circle� P and Q be analytic functionsin a neighborhood U of S�� such that P � �� � on S�� Let X be a RiemannSurface� �P and �Q � regular functions on X� and let � S� � X be ahomeomorphism such that P � �P � �� Q � �Q � � on S�� Then � can beextended as an analytic mapping of a certain neighborhood V � U of S� intoX� with the same property P � �P � �� Q � �Q � � in V �

Proof�P ��s� �� �� hence �P ����s�� �� �� so �P ��y� �� � in a neighborhood of ��s� on X�We dene ��x� in this neighborhood as y � ��x� � �P���P �x��� Locally �exists and is well�dened� Since these local extensions agree on S�� they infact agree and dene a required extension on a certain neighborhood of S���

Corollary ����� If in the Abel equation �A� on C

dy � y�dP � y�dQ or dy � y�dQ� y�dP

��

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P and Q are functions� satisfying all the properties ��� � ��� and the do�main D� provided by Wermer�s theorem� is simply�connected� then the Abelequation �A� has a center�

Proof�If P � �� � on S�� we may apply lemma ���� Then the Abel equation �A� isinduced by the analytic mapping � from the Abel Equation on X

dy � y�d �P � y�d �Q� � �A�

Since �P � �Q are analytic on a simply connected domain D bounded by ��S���the equation � �A� has a center along ��S��� and hence the equation �A� hasa center along S�� �

Notice� that this condition is a su�cient condition for center for Abelequation with arbitrary analytic coe�cients� But it is symmetric with respectto P and Q� although some of the center conditions for Abel equation areknown to be non�symmetric� Below we shall explain it for the case of P � Q� Laurent polynomials�

��� The degree of a rational mapping and an

image of a circle on a rational curve

Consider two rational functions P � Q� The map � � �P�Q� C � C � denesthe rational curve Y � f�P �t�� Q�t�� t � C g� Image of a circle S� under themap � is a closed curve on Y �

Theorem ����� Let P � Q be rational functions without poles on S�� s�t� atleast one of them has a pole inside S� and at least one of them has a poleoutside S� �for instance� Laurent polynomials�� Let ��S�� bound a compactdomain in Y � Then deg � � ��

Proof�Assume that deg � � �� Then consider a path � in C � joining two poles ofP and Q inside and outside of S� � for simplicity � and�� and intersectingS� only once at a regular point u � �� We can assume also that � does

��

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not contain preimages of double points in Y � So for any x � � there are noy �� x � C with ��x� � ��y��

����z�� tends to as z tends to � and to � so the image of ��z� un�der the map � C � Y can not stay inside a compact domain bounded by��S��� But it enters this domain� since u is a regular point of �� Then itmust intersect ��S�� at another point v �� u� and we get contradiction to thechoice of the path �� �

Example �� P �z� � z� Q�z� ��

z�

The rational curve Y is fxy � �g � C � � and the curve ��S�� does not bounda compact domain on it� The degree of the map �P�Q� is one� and it is ageneral situation for a map of degree one images of circles contracted topoles diverge on Y � see gure � with Sk � fjzj � kg� S�k � fjzj � �

kg

�k � N�� and Gk� G�k � their images on Y under � � �P�Q��

S

S

3

−1 [P,Q]

GG

1

3

GG G−1

−2 2

S−2

S2

S1

Example �� P �z� � z ��

z� Q�z� � z �

z�

The rational curve Y is fx � yg � C � the degree of the map � is �� and thecurve ��S�� bounds a compact domain on Y �in fact� S� � ��S�� � ���� ����See gure � for illustration images of the circles Sk cover Y twice� becausethey �have no space to diverge�� Arrows indicate directions of �motion� ofthe curves Sk and Gk as k decreases from � to ��

��

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S SS

S

S

G

G

32

1

-1

-2

2

G 1

G 3G-2

--1[ P , Q ]

��� Center conditions for the case of P Q �

Laurent polynomials

Theorem ���� If P and Q are Laurent polynomials� satisfying the condi�tion Z

jzj��

P �z�kQ�z�ndP � �

for all pairs �k�n� of nonnegative integers� and P� Q together separate pointson S�� then P and Q can be represented in the form of Laurent Polynomialcomposition� and hence the equation �A� has a center�

Remark� We believe that for the case of Laurent polynomials the Wermerstheorem remains valid without the assumption that P and Q together sep�arate points on S�� We hope to present a proof of this fact together witha detailed investigation of Wermers surface for the case of P � Q � Laurentpolynomials�

Proof�If both P and Q are algebraic polynomials in z� P and Q are represented asa composition with z� so we have a center�Similarly� if both P and Q are algebraic polynomials in �

z� P and Q are

represented as a composition with �z� so we have a center� Otherwise P and

Q have one pole inside �origin z � �� and one pole outside �innity� of S��Obviously P � Q are analytic in a neighborhood of S� and take only nitely

many values more than once on S��

��

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Next� we always can assume that P ��z� �� � for all jzj � �� Indeed� �A�after the change of variables z � u became

dy

du� d #P �u�y� � d #Q�u�y�� � #A�

where #P �u� � P �u�� #Q�u� � Q�u�� both �A� and � #A� have centersimultaneously� But as P � has only nite number of zeroes on C � by rescalingz �� z� which does not change a center for �A�� we can assure that there areno zeroes of #P � on the circle S�� For example� P �z� � z � �

zhas zeroes of P �

on S�� but #P �z� � �z � �zhas not� Under this change of variables the circle

jzj � � goes to juj � � but by Cauchy theorem for Laurent polynomialsintegrals along these circles coincide�Hence by Wermers theorem there exists a surface X and a homeomor�

phism � S� � X� such that ��S�� bounds a compact domain D on X andthere exists functions �P � �Q analytic inside D��Remind that we have a map � � �P�Q� C � Y � f�P �t�� Q�t�� t � C g�

Lemma ���� If the curve ��S�� bounds a compact domain on X� then thecurve ��S�� bounds a compact domain on a rational curve Y �

Proof�By lemma ��� �P�Q� � � is an analytic mapping� dened in a neighborhoodof S�� which coincides there with � � �P�Q�� But hence � �P � �Q� X � C

maps a neighborhood of ��S�� into Y � C � � By analytic continuation� � �P � �Q�maps X into Y � Hence � �P � �Q� maps a compact domain D inside ��S�� ontoY � and the image of a compact domain under continuous mapping is com�pact� Hence ��S�� is contained in a compact � �P� �Q��D�� Now one can easilyshow that in fact ��S�� bounds a compact domain in Y � �

Proof of theorem �� continue��By theorem ��� deg�P�Q� � �� hence P and Q can be represented as a com�position P � �P �w�� Q � �Q�w�� If �P and �Q are usual algebraic polynomials�we are done�If not� then P �z� � �P �zk�� Q�z� � �Q�zk� for Laurent polynomials �P �

�Q� But then on the rational curve Y we get �P�Q��S�� � � �P � �Q��S��� so� �P� �Q��S�� bounds a compact domain on Y � f�P �t�� Q�t�� t � C g �

��

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f� �P �t�� �Q�t�� t � C g� Therefore by theorem ��� deg� �P� �Q� � �� so they arerepresented as a composition�If we again obtain their representation as a composition of Laurent poly�

nomials with zn� we repeat our considerations� and nally we are left with

the composition �P � ��P � �w�� �Q � ��Q� �w� with �w � Laurent polynomial� ��P

and ��Q � algebraic polynomials� It gives us the composition P � ��P � �w�zN ���

Q � ��Q� �w�zN ��� and we are done� �

��

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Chapter �

Generalized Moments andCenter Conditions for Abeldi�erential equation withelliptic p� q

��� Introduction

In Chapter � we considered Generalized moments for rational functions Pand Q and have shown that vanishing of all generalized moments impliescomposition representability of P and Q and hence center for Abel equation�A�

dy � y�dP � y�dQ� �A�

In the proof we essentially used the fact that �P�Q��C � is a rational curve�i�e� algebraic curve of genus ��Suggested generalization is to consider the same question for Laurent

series instead of Laurent polynomials� The rst nontrivial case of Laurentseries in our �algebraic� context is the case of elliptic functions� For P � Q �elliptic functions we show that all the generalized moments

RP iQjdP along a

small curve around the origin vanish� But at the same time we demonstratethat in general Abel equation �A� with P � Q � elliptic functions does nothave a center� in spite of the fact that all the generalized moments on S�

vanish� So an analog of the theorem from �BY�� for innite Laurent series

��

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does not hold�It can be explained by the fact that elliptic curve X � �P�Q��C � is an

algebraic curve of genus � and hence it is homologically nontrivial� S� isembedded by the map �P�Q� into this curve in a homologically trivial way�and hence all the moments on S� vanish� But the domain� bounded by�P�Q��S�� is not simply�connected� Hence multiple integrals along this curvedo not vanish� and �A� does not have center�We start from the simplest case when P is the Weierstrass ��function� Q

is its derivative� In this case the surface �P�Q��C � is a torus with small circlearound the origin dividing it into two parts compact topologically nontrivialpart� on which � and �� are regular �have no poles�� and the second part wichis topologically equivalent to C without origin� but � and �� have a pole atzero�

picture

��� Generalized Moments on Elliptic Curves

As it was shown above �Chapter ��� the study of the center problem for theAbel equation leads naturally to the �moment�like� expressions of the formRP i dQ and

RQi dP � and similar more complicated ones�

All these expressions naturally appear as a part of the expressions for theTaylor coe�cients of the Poincare mapping given in Chapter � above�Transforming these coe�cients via integration by parts� one obtains ex�

pressions of the formRP iQj dP and

RP iQj dQ� and the expressions

mi�i�in�x� �

Z x

P i��x��q�x��

Z x�

P i��x��q�x�� � � �

Z xn��

P in�xn�q�xn� dxn � � � dx��

where p�x� � dP �x�dx� q�x� � dQ�x�dx �introduced by J�P� Francoise��

Expressions of this form� considered on complex curves of genus greaterthan zero� lead to multivalued functions� Hence we generalize denition of

��

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moments to this case

De�nition ����� For the two multivalued functions P � Q on a RiemannSurface X we de�ne generalized moments mi�j and m

�i�j as integrals along

a given curve � ofdmi�j � P iQjdQ �i� j � �� �����

dm�i�j � P iQjdP �i� j � �� �����

We normalize mi�j and m�i�j� assuming that all of them vanish at a �xed point

a� Here P and Q are analytically continued along the curve � from the pointa�

Remark ������ Let us notice� that

dm�i�j � Qjd

�P i��

i� �

�� d

�QjP i��

i � �

��P i��

i � �dQj � d

�QjP i��

i� �

�� j

i� �dmi���j��

Below we shall use this relation in computations of mi�j�

Following �BFY����� we can consider a problem of vanishing �or non�ramication� of generalized moments

Moment center problem� Find conditions on P � Q under which all themoments mk�n�x� for a certain set of indices �i� j� are univalued functionsalong a given curve � �and if � is not closed with end points a and b� then aand b are conjugate� i�e� mi�j�a� � mi�j�b� under analytic continuation along���

Below we present some rather preliminary computations in this directionfor elliptic functions�

Theorem ����� Let P � Q be a pair of elliptic functions with the only poleat zero� Then for all nonnegative k� n and all su�ciently small � �inside theparallelogramm of periods� the generalized momentsZ

juj�

P �u�kQ�u�ndP �����

vanish�

��

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Proof� The proof follows from the fact� that for su�ciently small � �smallerthan min���� ���� elliptic functions P and Q do not have poles �in the par�allelogramm of periods� outside the circle juj � �� Hence� the circle juj � �bounds �on the elliptic curve� a compact domain� on which the integrand isregular� �

Remark� We may show this fact� using combinatorial arguments� Remindsome basic notions from the theory of Weierstrass ��function� In local coor�dinates z around zero on C it can be dened by the formula

��z� ��

z��

�Xi��

ci z�i���

where coe�cients ci are computed by recursion

ci � �

Pi��j�� cj ci�j

�i� ����i� �� �

Here the rst coe�cients are dened as c� � g���� c� � g���� where g�and g� depend only on periods �

g� � ��X �

��� g� � ���

X �

���

where summation is taken over all non�zero peiods of ��The following combinatorial formulas hold

����� � ��� � g��� g�� ��� � ��� � g���

Now the proof of the theorem ����� follows from the fact that any in�tegral of the form ����� can be presented as a sum of integrals of the formRjuj�

��u�k��u��du andRjuj�

�k�u�du� Both integrals are zeros the rst be�

ing the increment of �k��

�� along juj � �� while the second contains only evendegrees of z and hence does not have residue inside juj � ��

Let us notice� that from remark ����� follows that generalized momentsRjuj�

P �u�kQ�u�ndP andRjuj�

P �u�kQ�u�ndQ are in equal� asRd�QjP i��

i��

��

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��

Next question we are interested in is computation of generalized momentsalong periods of elliptic integrals�

Theorem ����� Let � be Weierstrass ��function� ��� �� be periods� ThenZ�i

��u�n���u��k��du � ��

for all k and n�

Proof�R�i�n���k��du �

R�i�n���� � g��� g��

k��du �R�iF �����du � �� �

WILL BE ADDED

Computations forR�i��u�n���u��k� It is known that in general generalized

moments along period is equal to Z�i

� A�i �B�i�

where �i �R�i��u�du� well nd these A and B�

��� Abel equation with coe cients�elliptic func�

tions

As we saw in the previous section� all the generalized momentsRjuj�

P �u�kQ�u�ndP

for elliptic functions P � Q with the only pole at the origin vanish� For ra�tional functions that would imply center �under some additional restrictionson P and Q�� For elliptic functions the similar statement does not hold� Al�though for some elliptic P � Q the equation �A� may have center �say� P � Q��in general �A� does not have a center� In particular� the following theoremholds

Theorem ����� Let � be Weierstrass ��function� The equations

dy � y�d�� y�d�� and �����

��

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dy � y�d�� � y�d� �����

ramify along any small circle around zero�

Proof� By theorem ������ �Chapter �� in order for Abel di�erential equa�tion �A� to have center all the functions �k�z� in the expansion of PoincareReturn Map must be univalued functions of z� Substituting nite expansion�for su�ciently many ci to garantee all the necessary terms for all negativepowers of z� for ��z� into formulas of Chapter � for �k�z�� we nd that forAbel equation ����� ��z� has logarithmic term

�����g�� � ��g��� ln z� which is

the determinant of the Weierstrass function � and hence nonzero� i�e� ��z�is non�univalued function�

For Abel equation ����� all terms up to ��z� are univalued functions�but � �z� has logarithmic term

������g�� � ��g�����g� � ��� ln z� which can be

zero only for g� � ���� However� if g� � ���� ����z� has logarithmic term��

���g��g

�� � ��g��� ln z� and it is nonzero� �

WILL BE ADDED

Remark �� Well add handwriting proof of this fact� Also explanation�why ����� and ����� have di�erent �k vanishing �because of �

���

Remark �� Another remark is that in contrast with the notions of centerand conjugate points for the Abel equation on a plane� where all the deni�tions depend on homotopy class of the path �� here we meet with homologyclass of �� Here we integrate the closed forms P iQj dP � P iQj dQ severaltimes�

��

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REFERENCES�

�A� M�A�M� Alwash� On a condition for a centre of cubic non�autonomousequations� Proceedings of the Royal Society of Edinburgh� ���A ������ � �������

�AL� M�A�M� Alwash� N�G� Lloyd� Non�autonomous equations related topolynomial two�dimensional systems� Proceedings of the Royal Society of Ed�inburgh� ���A ������� �������

�B�� M� Blinov� Some computations around the center problem� related tothe algebra of univariate polynomials� M�Sc� Thesis� Weizmann Institute ofScience� ����

�BFY�� M� Briskin� J��P� Fran%coise� Y� Yomdin� The Bautin ideal of theAbel equation� Nonlinearity� �� ������� No �� �������

�BFY�� M� Briskin� J��P� Fran%coise� Y� Yomdin� Center conditions� com�positions of polynomials and moments on algebraic curves� Ergodic TheoryDynam� Systems �� ������� no� �� ���������

�BFY�� M� Briskin� J��P� Fran%coise� Y� Yomdin� Center conditions IIParametric and model center problems � Israel J� Math� ��� ������� ������

�BFY�� M� Briskin� J��P� Fran%coise� Y� Yomdin� Center conditions IIIParametric and model center problems � Israel J� Math� ��� ������� �������

�BFY�� M� Briskin� J��P� Fran%coise� Y� Yomdin� Generalized Moments�Center � Focus Conditions and Composition of Polynomials � Operator the�ory� system theory and related topics �Beer�Sheva&Rehovot� ������ ��������Oper� Theory Adv� Appl�� ���� Birkhuser� Basel� �����

�BY�� M� Blinov� Y� Yomdin� Generalized center conditions and multi�plicities for the polynomial Abel equations of small degrees� Nonlinearity� ��������� ���������

��

Page 87: University of Connecticut · Ac kno wledgmen ts I w ould lik e to thank m y advisor Professor Y osef omdin for his great sup ervision constan t supp ort and encouragemen t for guiding

�BY�� M� Blinov� Y� Yomdin � Center and Composition Conditions forAbel Dierential Equation� and rational curves� Qualitative Theory of Dy�namical Systems� � ������� �������

�BY�� M� Blinov� Y� Yomdin � Center�Focus and Composition Conditionfor Abel Dierential Equations on Riemann Surfaces and associated Gener�alized Moments�� in preparation

�Che�� L� Cherkas� Number of limit cycles of an autonomous second�ordersystem � Di�erentsialnye uravneniya �� ������� No��� �������

�Che�� L� Cherkas� On the conditions for a center for certain equationsof the form yy� � P �x� � Q�x�y � R�x�y�� Di�erential equations � ����������������

�Chr�� C� Christopher� An algebraic approach to the Classi�cation of Cen�ters in Polynomial Lienard Systems� J� Math� Ann� Appl� ��� ������� �������

�Chr�� C� Christopher� Abel equations composition conjectures and themodel problem� Bull� Lond� Math� Soc� �� No� �� ������� ������

�Dev�� J�Devlin� Word problems related to periodic solutions of a nonau�tonomous system� Math� Proc� Camb� Phi� Soc� ������� ���� �������

�Dev�� J�Devlin�Word problems related to derivatives of the displacementmap� Math� Proc� Camb� Phi� Soc� ������� ���� �������

�ER� A� eremenko� L�Rubel� The arithmetic of intire functions under com�position� Advances in Math�� �� ������� �������

�F� J��P� Fran%coise� Private communications

�GL� A� Gasull� J� Llibre� Limit cycles for a class of Abel equations � SIAMJ�Math� Anal� �� ������� No��� ���������

��

Page 88: University of Connecticut · Ac kno wledgmen ts I w ould lik e to thank m y advisor Professor Y osef omdin for his great sup ervision constan t supp ort and encouragemen t for guiding

�L� A� Lins Neto� On the number of solutions of the equationdx

dt�

nXj��

aj�t�xj� � � t � �� for which x��� � x���� Inventiones math� �� �������

�����

�N� N� Roytvarf� �Generalized moments� composition of polynomials andBernstein classes� in �Entire functions in modern analysis� B�Ya� Levinmemorial volume�� Isr� Math� Conf� Proc� ��� ������� ������

�Pa�� F� Pakovich� A counterexample to the �Composition Conjecture��to appear in Proc� AMS

�Pa�� F� Pakovich� On the polynomial moment problem� preprint

�R� J� Ritt� Prime and composite polynomials� Trans� AMS� �� ������������

�Sha� I� Shafarevich� Basic algebraic geometry� Springer�Verlag� ����

�Sch� D� Schlomiuk� Algebraic particular integrals� integrability and theproblem of the center � Trans� AMS ��� ������� No��� �������

�W�� J� Wermer� Function Rings and Riemann Surfaces� Annals of Math��� ������� �����

�W�� J� Wermer� The hull of a curve in Cn� Annals of Math� �� ��������������

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