Center and Comp osition conditions for Ab el

Equation

Thesis for the degree of Do ctor of Philosophy

by

Michael Blinov

Under the Sup ervision of professor

Yosef Yomdin

Department of Theoretical Mathematics

The Weizmann Institute of Science

Submitted to the Feinb erg Graduate Scho ol of

the Weizmann Institute of Science

Rehovot Israel

June

Acknowledgments

I would like to thank my advisor Professor Yosef Yomdin for his great

sup ervision constant supp ort and encouragement for guiding and stimulat

ing my mathematical work I should also mention that my trips to scien

tic meetings were supp orted from Y Yomdins Minerva Science Foundation

grant

I am grateful to JP Francoise F Pakovich E Shulman S Yakovenko

and V Katsnelson for interesting discussions

I thank Carla Scapinello Uruguay Jonatan Gutman Israel Michael

Kiermaier Germany and Oleg Lavrovsky Canada for their results obtained

during summer pro jects that I sup ervised

I am also grateful to my mother for her patience supp ort and lo ve I wish

to thank my mathematical teacher V Sap ozhnikov who gave the initial push

to my mathematical research

Finally I wish to thank the following for their friendship and in some

cases love Lili Eugene Bom Elad Younghee Ira Rimma Yulia Felix

Eduard Lena Dima Oksana Olga Alex and many more

Abstract

We consider the real vector eld f x y gx y on the real plane IR

This vector eld describ es the dynamical system

x f x y

y g x y

A classical CenterFo cus problem whichwas stated byHPoincare in s

is nd conditions on f g under which all tra jectories of this dynamical sys

tem are closed curves around some point This situation is called a center

In some cases this problem is reduced to the problem of nding conditions

for solutions of Ab el dierential equation

p q

to be p erio dic on the interval

In the thesis various sp ecial cases of the CenterFocus problem for Ab el

Equation are investigated through their relation to Comp osition algebra of

p olynomials and rational functions to Generalized Moments and to Algebraic

Geometry of Plane Curves

Contents

Intro duction

Center and Comp osition Conditions for vector elds on the

plane

Intro duction

Cherkas transform

General transform from a plane vector eld to a rst order

nonautonomous dierential equation

Comp osition conditions for Symmetric and Hamiltonian com

ponents in general case

Hamiltonian system

Symmetric comp onent

Comp osition and center conditions for quadratic planar systems

The comp osition condition for the Hamiltonian com

p onent

Comp osition condition for the symmetric comp onent

Noncomp osition comp onents for quadratic systems

Cubic p olynomial vector elds

Comp osition conditions for the Hamiltonian comp o

H

nent C

condition for the comp onent C do es Comp osition

not hold

Comp osition conditions for the Symmetric comp onent

R

C

Poincare return map for Ab el dierential equation and com

putations of center conditions

Intro duction

Poincare return map for Ab el equation and recurrence relations

Generators of ideals I

k

Center and Comp osition Conditions for Ab el dierential equa

tions with p q trigonometric p olynomials of small degrees

Intro duction

Computational approach

Center conditions for p q of small degrees

Center conditions for the case when either p or q is of

the degree zero

Center conditions for the case deg p deg q

Center conditions for the case deg p deg q

Center conditions for the case deg p deg q

Center conditions for the case deg p deg q

Center and Comp osition Conditions for Ab el dierential equa

tions with p q algebraic p olynomials of small degrees

Intro duction

Comp osition conjecture for multiple zero es and its status for

small degrees of p and q

Description of a center set for p q of small degrees

Comp osition conjecture for some sp ecial families of p olynomials

Riemann Surface approach to the Center problem for Ab el

equation

Intro duction

Ab el equation on Riemann Surfaces

Example Polynomial Comp osition Conjecture in the

case X C P and Q p olynomials in x

Example Laurent Comp osition Condition in the case

X a neighb orho o d of the unit circle S fjxj g

C P and Q Laurent series convergent on X

Poincare mapping

Factorization comp osition for the case X C P and Q

rational functions

Intro duction

Comp osition and rational curves

Structure of comp osition in the case X C P and Q poly

nomials

Structure of comp osition in the case X a neighborhood of

the unit circle on C P and Q Laurent p olynomials

Moments of P Q on S and Center Conditions for Ab el equa

tion with rational p q

Intro duction

Sucient center condition for Ab el equation with analytic p q

The degree of a rational mapping and an image of a circle on

a rational curve

Center conditions for the case of P Q Laurent p olynomials

Generalized Moments and Center Conditions for Ab el dif

ferential equation with elliptic p q

Intro duction

Generalized Moments on Elliptic Curves

Ab el equation with co ecientselliptic functions

Chapter

Intro duction

Let F x y Gx y b e algebraic p olynomials in x y without free and linear

terms Consider the system of dierential equations

x y F x y

y x Gx y

One can reduce the system with homogeneous F G of degree d to Ab el

dierential equation

p q

where p q are p olynomials in sin cos of degrees dep ending only on

d Then has a center if and only if has all the solutions p erio dic

satisfying y y on ie solutions

The classical center problem as stated for is to nd conditions on p

and q such that has a center We shall call it Center Condition

The following simple sucient condition was intro duced in AL Let

w C be a function such that w w Let

p pw w

q qw w

Then all the solutions of have the form w hence they

satisfy the condition

We shall call the representation Comp osition Condition on p q

The comp osition condition is sucient but not necessary for the center

The main sub ject of this research is the investigation of relations b etween

center and comp osition conditions

In the next chapter I intro duce statement of center problem for vector

elds on the plane Reduction of the problem to the problem for Ab el dif

ferential equation Cherkas transform is demonstrated and an attempt is

done to generalize it Center and comp osition conditions are investigated for

dynamical systems on the plane with F G homogeneous p olynomials of

degrees and We show which center comp onents can b e represented as a

comp osition and which can not

In the third chapter I intro duce Poincare return map for Ab el dierential

equation Recurrence relations and corresp onding ideals are studied and

generators for these ideals are found

In the fourth chapter of the thesis Ab el dierential equation is studied

with p q trigonometric p olynomials of small degrees up to It is shown

that in these cases center and comp osition conditions coincide although it is

known that for greater degrees of p q some center conditions are not given

by comp osition

In the fth chapter of the thesis center and comp osition conditions are

studied for Ab el dierential equation with p q algebraic p olynomials of

small degrees It was shown that these conditions coincide Some generaliza

tion of center problem are intro duced Weshow also some sp ecial families of

p olynomials for which equivalence of the center and the comp osition condi

tions can be easily sho wn

In the sixth chapter Riemann Surface approachtotheCenter problem for

Ab el equation is discussed This is a convenient general setting where Ab el

Equation is considered on a given Riemann Surface The notions of center

and comp osition are generalized accordingly

In the seventh chapter we study comp osition for rational functions We

establish some facts ab out structure of comp osition for algebraic p olynomials

and Laurent p olynomials

In the eighth chapter center problem is studied for Ab el dierential equa

tion with rational functions p q One can rewrite the dierential equation

i

in a complex form expressing sin and cos through z e ie

z z

cos

z z

sin

i

y

so one obtains p and q in the form of Laurent p olynomials in z and Ab el

dierential equation is

dy

pz y q z y

dz

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

R R

dition on P p and Q q namely vanishing of generalized moments

R

i j

P Q dP implies center

jz j

Finally in the last chapter of the thesis Riemannsurface approach is

extended for the case of elliptic functions Sp ecically for elliptic functions

R

i j

P Q dP P Q it is shown that in general the condition do es not

jz j

imply center The dierence between the Laurent p olynomials and elliptic

functions P Q is shown to be in the top ology of the complex curve Y

fP z Qz z C g C In the rst case this curve is rational and in

the second case it is top ologically a torus The onedimensional homology of

the torus is resp onsible for ramication of the solutions of the Ab el equation

R

i j

in spite of vanishing of all the moments P Q dP

jz j

Chapter

Center and Comp osition

Conditions for vector elds on

the plane

Intro duction

The following formulation of the center problem see eg Sch for a general

discussion of the classical center problem is considered Let F x y Gx y

be algebraic homogeneous p olynomials in x y of degree d Consider the

system of dierential equations

x y F x y

y x Gx y

A solution xt y t of is said to be closed if it is dened in the

interval t and x xt y y t The system is said to

have a center at if all the solutions around zero are closed Then the

general problem is under what conditions on F G the system has a

center at zero

It was shown in Ch that one can reduce the system with homoge

neous F G of degree d to Ab el equation

p q

where p q are p olynomials in sin cos of degrees d and d

resp ectively Then has a center if and only if has all the solutions

p erio dic on ie all the solutions r r with suciently small

satisfy Conditions on co ecients of p q under which has

all the solutions p erio dic are called center conditions for Ab el equation

Following AL in BFYBFY the following simple sucient condition

Z Z

was stated Let P psds Q q sds Then

Comp osition Condition The sucient condition for the existenceofthe

center for is the representability of P Q as a composition of al

gebraic polynomials with trigonometric polynomial ie there exist a trigono

metric polynomial W polynomial of cos sin st W W

and algebraic polynomials P Q without free terms st P P W

QQW

The suciency follows from the fact that if P P W Q QW then

the solution of W and as W W

The full center conditions are known yet only for homogeneous p erturba

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 comp onents are

given in fact by comp osition section

explicitly write down comp osition representation for Hamiltonian and

Symmetric comp onents in quadratic section and cubic section

cases

show by counterexample that other known comp onents for quadratic

and cubic homogeneous vector elds are in general unrepresentable as

a comp osition

Cherkas transform

We shall show how in a sp ecial case of systems the problem of nding

center conditions on F x y Gx y can be reduced to the center problem

for Ab el equation

Write in p olar co ordinates x r cos y r sin We obtain

x r cos r sin r sin F r cos r sin

r cos r sin y r sin r cos r cos F

Multiplying the rst equation by cos the second by sin and adding

we get

r F r cos r sin cos Gr cos r sin sin

d d

r F cos sin cos Gcos sin sin r f

In a similar way

d d

r F cos sin sin Gcos sin cos r g

Finallywe get

d

dr r r f

d

d r g

Then we apply a transformation suggested by L Cherkas in Che

d

r

d

r g

d

r

g

Notice that for d this transformation and its inverse are regular at

r For d Cherkas transformation considered over complex

numbers ramies at zero but for small real r it is monotone

d

Denote r g by A We have

dr dr d

d d d d

A r g r g d r d r

d A d d

d d

r r

d

d f g r d f g

A A

d f g d f g

Thus we obtain

d

p q

d

where p d f g and q d f g are p olynomials

in sin cos of degrees d and d resp ectively

Clearlythe tra jectory of near the origin is closed if and only if the

corresp onding solution of satises and hence is p erio dic

Therefore the centerfo cus problem for is translated into the problem

of nding conditions on p and q under which all the solutions of are

p erio dic

General transform from a plane vector

eld to a rst order nonautonomous dif

ferential equation

In this section a transform from a p olynomial plane vector eld to a rst order

nonautonomous dierential equation is illustrated Let F x y Gx y be

arbitrary p olynomials of degree grater than We can consider any p olyno

mials F x y Gxy as a sum of homogeneous p olynomials F G of degree

d d

d

F F F F

k

G G G G

k

In a similar way to the homogeneous case we obtain

k

r r f r f r f dr

k

k

d rg r g r g

k

When r is suciently small we can write it as

X

dr

k i k

rg r g r g r f r f r f

k k

d

i

After multiplication we obtain formal innite p ower series

X

i

r a r

i

i

where a are p olynomials in sin cos

i

Nowwe can apply normal form technique eliminating one by one all the

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

X X

i i

uu u r r r r r r r a r a r

i i

i i

a u a u a u

So by cho osing a we get dierential equation with the right hand

side p ower series without u Continuing elimination we can kill all terms

k

u for k there are some obstructions to this pro cess which we dont

discuss now These initial computations suggest a way to transform any

p olynomial vector eld to a rst order nonautonomous dierential equation

Comp osition conditions for Symmetric and

Hamiltonian comp onents in general case

Hamiltonian system

We consider the system

H

x y x y

y

H

y x x y

x

with H x y homogeneous p olynomial in x y of degree d After trans

forming to p olar co ordinates we get

d

r r f

d

r g

with

H H

f cos sin cos cos sin sin

y x

dH

H cos sin hence f

d

H H

g cos sin sin cos sin cos

y x

Due to homogeneityofH g d H cos sin

Applying Cherkas transformation we obtain the Ab el equation

p q with p d f g q d f g

Integrating we get

P dH cos sin dH cos sin C dH cos sin C

dH cos sin

q d H cos sin

dt

hence

d

Q H cos sin C

and

d

P C P Q

d

where C is a constant

Symmetric comp onent

We consider the planar system

x y F x y

y x Gx y

As we can rotate our system around the origin it is enough to consider the

case of symmetry with resp ect to xaxis

F x y F x y

Gx y Gx y

In p olar co ordinates wegetf F cos sin cos Gcos sin sin

f g F cos sin sin Gcos sin cos g

After Cherkas transformations we get p d f g q

q are odd functions hence P and d f g hence p and

Q are even functions ie they are functions of cos

Below we write down explicit expressions for co ecients of such comp o

sitions for Hamiltonian and Symmetric cases for homogeneous p olynomial

planar systems of the second and the third degrees

Comp osition and center conditions for quadratic

planar systems

The system with homogeneous F x y Gx y of degree can be

rewritten in the complex notations Zoladek Zo with parameters

z iz Az Bzz C z

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

It is known see Zo that the comp onents of the center set in this case

are

LV

B LotkaVolterra comp onent Q

R

ImAB ImB C ImA C Symmetric comp onent Q

H

A B Hamiltonian comp onent Q

A B jC jjB j Darbu comp onent Q

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

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

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

The center conditions are as follow

LV

b b Q

a b a b

R

b c b c b b c b b c

Q

a c a c a a c a a c

H

a b a b Q

a b

a b

Q

c c b b

We shall nd the direct representations of Symmetric and Hamiltonian

comp onents as a comp osition and we shall show that in general case Lotka

Volterra and Darbu comp onents can not b e represented as a comp osition

The comp osition condition for the Hamiltonian

comp onent

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

Qx P x a c P x

Pro of Using the M athematica program here is the main function

Fxylambda x lambda y

Gxy lambda lambda x y

fGCosfiSinfiSin fi F Cos fi Sin fi C osf i

gGCosfiSinfiCos fi F Cos fi Sin fi S inf i

paTrigExpandfDgfi

qaTrigExpandfg

PTrigReduceIntegratepa fi x

QTrigReduceIntegrateqa fi x

we nd

P a c a cosxc cos xa sinx c sinx

a cosxa cosx Q a a c a c c

a c cos x a c cos x a c cos xa c cosx c cosx

c cos xa a sinx a c sinx a c sinx a c sinx

a c sinx c c sinx

Now to cho ose in the representation Qx P xP x we

compute P a c Q hence a c and

the identity

Qx P x a c P x

holds

Comp osition condition for the symmetric com

p onent

Lemma In Symmetric case there exists a composition with the polynomial

of the rst degree of the form W x cos x sin x where is a constant

which is found from the cubic equation involving coecients of the system

Pro of 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 b oth b and b

are zero es we are in LotkaVolterra case Then from the rst equation

a a b b

and from the t wo last equation we can nd either

b b c

or

b

b c b b c

b b b

p

Consider the rst case when b b Running M athematica we get

p p

b b cos xb sinxc sinx P

p

Q a b a c b c c a cos xb cosx

p p p

a c cos x b c cos x c cosx a sinx b sinx

p

a c sinx b c sinx

After some computations we obtain comp osition with the rst degree

p

p olynomial W xcosx sinx

p p p p

W x c W x c b W x P xc

c c b c b c c

Q c W W W W

p

p

a b a c b c c

a c b c W W a b

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

comp osition

Noncomp osition comp onents for quadratic sys

tems

One of ways to check if two p olynomials P x Qx have a common divisor

is to plot graphs of p olynomials P x P y and Qx Qy on the real

plane x y and to see if there is a common subgraph outside the diagonal

x y For real curves these subgraphs are real see ER for detail

We plot graphs for dierent numerical values in each of the four comp o

nents using the Matlab software

For LotkaVolterra comp onent we consider the numerical example with

parameters a a c c ie

F x y x y

Gx y x xy y

Darbu comp onent we consider the numerical example with parame For

ters a a c c ie

F x y x y

Gx y x xy y Plots of P(x)−P(y) (red) and Q(x)−Q(y) (green). 3

2

1

Y 0

−1

−2

−3 −3 −2 −1 0 1 2 3

X

Figure Graph for LotkaVolterra comp onent

Plots of P(x)−P(y) (red) and Q(x)−Q(y) (green). 3

2

1

Y 0

−1

−2

−3 −3 −2 −1 0 1 2 3

X

Figure Graph for Darbu comp onent

For LotkaVoltera and Darbu comp onents we see that there is no common

subgraph Although rigorous pro of would require discussion of computer pre

cision of Matlab software and graphical capabilities of hardware we assume

that visually observable distance b etween curves is greater than machine er

ror and there is no common subgraph These graphical counterexamples

prove that there is no comp osition for these two comp onents

Cubic p olynomial vector elds

The system with homogeneous F x y Gx y of degree can be

rewritten in the complex notations Zoladek Zo with parameters

z iz Dz Ez z Fzz Gz

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

It is known see Zo that the comp onents of the center set in this case

are

H

ReE D F C

R

ReE ImDF ReD G ReF G C

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

d e f g y e d f g xy

The center conditions are as follow

H

e d f d f C

e d f d f

R

d g d g d d g

C

f g f g f f g

e e

d f d f

C

g g f f

We shall nd the direct representations of Symmetric and Hamiltonian

comp onents as a comp osition and we shall show that in general case Darbu

comp onent can not be represented as a comp osition

Comp osition conditions for the Hamiltonian com

H

p onent C

Lemma In Hamiltonian case the system has the form

d g x e g x y 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

and there exists a composition of the form

P d e g P Q

Pro of After running M athematica we get

P d g d cos xg cos xd sinxg sinx

Q d d d e g d g e g g d e cosx

cos x e g cosx cos x d d g cosxd g cosx d

d g cosxd g cosxg cos xg cos xd e sinxd g sinx

d g sinxd d sinx e g sinx d g sinxd g sinx

g g sinx

Substituting x we get P d Q d e g hence

d e g Now we get

Q P d e g P

qed

So in Hamiltonian case b oth P and Q can b e expressed as a comp osition

of some usual p olynomials with a certain trigonometric p olynomial namely

P

Comp osition condition for the comp onent C

do es not hold

In usual notations on R this comp onent is given by the following system

x y f g x f g x y

f g y f g xy

y x f g x f g x y

f g xy f g y

with an additional relation on co ecients g g f f

We compute using program which utilizes only the rst four of the

equations We get

P f g f cosxg cos xf sinxg sinx

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

f g cosxf cos x f cosx f g cos xf g cos x

g cos x g cos x f g sinx f g sinxf f sinx

f g sinx f g sinx g g sinx

where there is one additional relation on co ecients

g g f f

After trying to get a comp osition with p olynomials of the rst sin x

cos second sin x cos x sin x cos x and

the forth simply P degrees we got contradiction

Comp osition conditions for the Symmetric com

R

p onent C

with a polyno Lemma In symmetric case there exists a composition either

mial of the rst order in sin x cos x or of the second order Namely

if d or d d there exists a composition with polynomial

W x cos x sin x where is dened from a cubic equation on

if d and d there exists a composition with W cos x

Pro of Here wehave the only one usable equation e and three nonlin

ear equations But we can solve the system of center conditions considering

separately 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

d f

f

d

into the third equation and we get instead of the third equation the following

one

f

d g d g d d g

d

We see that the third equation disapp eared 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 comp osition with

W x cosx sinx

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

d d

g g

d d

In this case we obtain the comp osition with W cos x sinx

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

Again there are two cases

a

d d

f g f g f f g

In this case we obtain a comp osition with W cosx sinx

where is found from the cubic equation

b d f g

P and Q are b oth functions of cos x

Chapter

Poincare return map for Ab el

dierential equation and

computations of center

conditions

Intro duction

We consider Ab el equation

y px y q x y

with p q any analytic functions We can x any pont a instead of and

ask the same question Under which conditions on p and q are all tra jectories

p erio dic ie any tra jectory starting at with the value y assumes the

same value y at a

In this chapter

e write down recurrence relations for co ecients of as in BFY w

Poincare return map

using these recurrence relations we compute several rst co ecients

manually and nd out generators of co ecient ideals

we describ e an algorithm to nd these generators using Mathematica

symb olic computer system

Poincare return map for Ab el equation

and recurrence relations

We maylookforsolutions of in the form of Poincare return map

X

k

y x y y v x y

k

k

where y y y is the nite set of the co ecients of p

q Shortly we will write v x

k

X

k

y a Then y ay a y y v ay and hence the condition

k

k

y is equivalent to v a for k

k

One can easily showby substitution of the expansion into the equa

tion that v x satisfy recurrence relations

k

v x

v x

v and

n

X X

xv xv x n v xpx v xv xq x v

i j k i j

n

ij n

ij k n

It was shown in BFY that in fact the recurrence relations can

be linearized ie the same ideals I fv v v gs are generated by

k k

f x xg where x satisfy linear recurrence relations

k k

x

x

and

n

xn xpx n xq x n

n n

n

which are much more convenient then

Direct computations including several integrations by part give the fol

lowing expressions for the rst p olynomials x solving the recurrence

k

Z Z

x x

relation Here we denote P x ptdt Qx q tdt

x P x

x P x Qx

Z

x

x P xP xQx q tP tdt

Z

x

x P x P xQx q tP tdt

Z

x

P x q tP tdt Q x

Z

x

x P xP xQxP x q tP tdt

Z Z

x x

Q xP x P x q tP tdt Qx q tP tdt

Z Z

x x

tdt q tP ptQ tdt

Z

P P Q P tq tdt

Z

q tP tdt P Q P

Z Z

P Q P tq t dt P q tP tdt

Z Z

P Q tptdt P tq tdt

Z

Q P Q P tptQ tdt

and similar much more longer expression was obtained for x

Generators of ideals I

k

Consequently we get the following set of generators for the ideals I k

k

I fP g

I fP Qg

Z

I fP Q qP g

Z Z

qP qP g I fP Q

Z Z Z

I fP Q qP qP qP pQ g

Z Z Z Z

pQ qP PpQ g I fP Q qP qP qP

Z Z Z Z

I fP Q qP qP qP pQ qP PpQ

Z Z

Q p P Qq P q Pqg qP

These generators were rst computed by Alwash and Lloyd in AL us

ing the nonlinear recurrence relation and recomputed now using the

recurrence relation

The problem to nd generators of ideals I was studied also from com

k

binatorial point of view by J Devlin Dev Dev We tried to write a

program to nd all the generators using symb olic computations with Math

ematica software

The diculty lies in that the general formula for cannot b e deduced

k

using the computer for each given p Resp ectivelywe can compute

k

q but we cannot compute the formula for a general symb olic 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 number

of integrals in the general formula as demonstrated in section We

attempted to write a program which could compute integrals in a symbolic

form using integration by parts This work was done together with Oleg

Lavrovsky working on this summer pro ject under my sup ervision

Mathematica is an integrated technical computing system that allows for

numerical and symbolic computations of high complexity However there

are no libraries that would facilitate manipulation of symb olic functions due

to the following two diculties

a p q are symb olic functions making the use of standard Integrate

and Differentiate functions of Mathematica inapplicable

b an interface is necessary for allowing the user to sp ecify the transformation

or exclusion of comp onents during the iteration because there is a certain

reasonable limit on the complexity of transformations that we dene Un

ending switching lo ops often rise between two or three p ossible formats of

expressions as the computer is unable to decide on a result ab ove

To overcome the rst diculty we created a new set of functions The

function i is used to representanintegral replacing the standard function

Integrateanddv corresp ondingly the default function Differentiate

We compute using recursive function

fnfnniExpan dp fnniExpandq fn

with initial conditions f f

We start with the functions p and q as arbitrary symbols we intro duce

the complimentary set P and Q so the output fn will contain only p q

P Q and integration op eration i

dvP p

dvQ q

ip P

iq Q

Within the Mathematica kernel expressions of fn are traced back

until the rst dened values f and f and each successive expression

calculated from therein and stored

The main set of transformations was dened in this fashion

ix y ix iy

dvx y dvx dvy

dvx y dvx y dvy x

idv xx dvixx

i p Pnn Pn

etc

Using these rules if Mathematica encounters an expression such as

ip q ip P

it transforms it in the following way

ip iq ip PPip iq ip PP

PQ PPP

This continues until the rule is no longer applicable To overcome the

second diculty we dene integration by parts as

ip B P B iP dvB Test

where B is any expression Test represents a pro cedure to display a

prompt for accepting or den ying transforming of the given expression Note

that it was critical that the ordering of these expressions for execution was

carefully determined during analysis of such problems so that the passing

of conversions should follow eective sequences As we can apply many dif

ferent integration by parts taking dierent expressions as derivatives under

integration sign this step requires some complicated testing pro cedures

A set of functions was also added that allow the user to cho ose terms

to break down further again progressing the formula into a more prefer

able form This was based on separately transforming the chosen terms and

inserting the result into the general expression then simplifying

The develop ed algorithm proved to be successful in the evaluation of

in the extent of comparison with currently known expressions up to

k

k and had the required integral complexity in higher expressions For

comparison the manual computation of takes many hours program

compute it during several minutes The work on the algorithm is not nished

as for with larger k wehave ac hoice of many expressions to integrate

k

by parts and Test pro cedure must b e done much more complicated

Chapter

Center and Comp osition

Conditions for Ab el dierential

equations with p q

trigonometric p olynomials of

small degrees

Intro duction

As it was shown in Ch starting from homogeneous p erturbation F x y

d

G x y of degree d we obtain Ab el equation with p q homogeneous

d

p olynomials in sin x cos x of degrees d d resp ectively It means that

the lowest degrees of trigonometric p olynomials in related to dynamical

system are and resp ectively The center conditions for them are

known and they are not restricted to the comp osition conditions

One can consider Ab el equation

pxy q xy y

with px q x arbitrary trigonometric p olynomials ie p olynomials of any

degree As b efore Center Condition for Ab el equation is p erio dicityof

solutions y x between two sp ecied points and ie y y for

all solutions y x starting at with suciently small initial value y

As b efore comp osition condition in the form P xP W x Qx

QW x for some function W x with W W is sucient condi

tion for the center

The question when center and comp osition conditions coincide can be

stated for any functions p q For px q x algebraic p olynomials in x of

n n

the form a x a x a this question was studied in BFY

n n

BY and the following Comp osition Conjecture was stated for alge

braic polynomials p q composition condition is not only sucient but also

necessary for the center The computational verication for p q algebraic

p olynomials of small degrees was done in BY

We consider trigonometric p olynomials in a general form

deg p

X

p sink cos k

k k

k

Such representation is unique since the functions cos sin cos sin

form a basis in the space of all trigonometric p olynomials As the degree of

trigonometric p olynomial the maximal d was considered such that

d

or is not equal to zero As free term we consider

d

In this chapter we show that for small degrees of trigonometric p olynomi

als p q up to center condition implies comp osition condition All these

comp osition conditions will b e directly written down Part of computations

was done together with M Kiermaier working on this summer pro ject under

my sup ervision

Computational approach

We proved the following theorem

composition condition is not only Theorem For p q up to degrees

sucient but necessary for the center

To prove it we will compute subsequently co ecients x of Poincare

k

return map

X

k

y x y y x y

k

k

and after substitution x we will lo ok for conditions on co ecients

under which The integration of trigonometric functions

i i k

works very slow in Mathematica so to increase the sp eed of computations

a function integ was intro duced which replaced the standard Integrate

pro cedure of Mathematica

RecursionLimit

integy zxintegyxintegzx

integc yxc integyx FreeQcx

integcxc x FreeQcx

integxnxxnn FreeQnx n

integSina xxCosa xa a FreeQax

integCosa xx Sina xa FreeQax

integSinxxCosx

integCosxx Sinx

Then weuseTrigReduce to reduce trigonometric expressions to the form

where integ can b e applied

psixi

TrigReduceintegTrigReduceipsixipxipsixiqxx

After computing rst several equations we prove that these

k

conditions imply comp osition representabilityofP Q and hence im

ply center

Let us x notations we work with Let us notice that integrating

Z Z

wrt and taking into consideration cos tdt we sin tdt

got that p q do not have free terms ie

deg p

X

p cos sin

i i

i

deg q

X

q cos sin

i i

i

which corresp onds to the following P Q

deg p

X

P sin cos

i i i

i

deg q

X

Q sin cos

i i i

i

Notice that in contrast to the p olynomial case the degree of trigonomet

ric p olynomial do es not change after integration or dierentiation

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 b ecomes

y q xy

a

Lemma The condition Q is necessary and sucient for the

existence of a center

Pro of This equation can b e solved explicitly

y

q x QxC onst Qx

y y y

Now its obvious that the center substituting x we get C onst

y

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 denes center Formal ly speaking there exists a

espectively composition with polynomial q p r

Center conditions for the case deg p deg q

Lemma For deg p degq center conditions coincide with

the composition conditions ie there exists center i P kQ

Pro of We get

P sin cos

Q sin cos

After computing we get

only if therefore this condition

is necessary for center Hence k k for some k hence

P kQ

Center conditions for the case deg p deg q

Lemma For deg p deg q center conditions coincide

with the composition conditions There are three center components for this

case

A

corresponding to the composition with W cos

B

corresponding to the composition with W sin

C

corresponding to the composition

P P Q

Pro of We get

P sin cos

Q sin cos sin cos

After computing we get

We assume that

if either

Case and

or

Case In this case is not a solution b ecause we get

deg p so we left with the only p ossibility

Consider the rst case Computing under the written relation

we get and

Solving the equation we get two cases

and Case

Case

Case Computing we get

for all the three cases Hence we have obtained three sets of con

ditions on co ecients which are suspicious of b eing center conditions Lets

analyze them

Case

In this case we get P cos

Q cos cos cos cos

cos cos cos cos

cos

So there is a comp osition with W cos P W W and

QW W W so this condition is a comp osition con

dition and therefore this condition is sucient 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 comp ositions with W sin this

condition is comp osition condition and hence it is sucient for the existence

ofacenter

Case

In this case we get

P sin cos

Q sin cos sin cos

Lets lo ok for the comp osition condition QaP bP

Substituting P and Q into this equation we get

a a cosxb a sinx a sinx

a b a b a cosx

Solving this system we obtain

a b

For this values of a and b we veried that aP bP Q

and therefore in the case center condition coincide with the comp osition

condition

Center conditions for the case deg p deg q

Lemma For deg p deg q center conditions coincide

with the composition conditions There are four center components for this

case

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 and we get that these representation coincides with the

k k

composition 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 wel l

Pro of We get

P sin cos sin cos

Q sin cos

After computing we get

We assume that and

i at least one of the three conditions is satised

Case

Case

Now we compute for these cases

so we obtain In the case we get

the following cases

Case

Here

sin cos sin cos P

Q sin cos

solving the equation P x aQx bQx wrto a b we found a

b and for these values of a bPx aQx bQx

So this case is comp osition case

Case Then

P cos cos cos l cos

Q cos so we get comp osition with W cos

hence it is center condition

In the case we get

m and

hence from follows From the third

solution follows p hence this case is imp ossible and we are left with

the two cases

Case and in this case also and we

get

P sin cos sin sin Q sin

so P and Q are comp ositions with W sin

Case and then from follo ws

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

generality we can shift co ecients p and q of the Ab el Without loss of

equation on the angle

px px

q x qx

Then

Z Z

P p xdx px dx P x P

new new

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 sin xsin sin xcos sin xsin sin xcos 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 co ecient b eing equal to zero let

then using rescaling we get b oth of them can not be zero es since

the degree of P is

Lemma For

P sin cos cos

Q sin cos sin cos

the necessary and sucient conditions for the center is either P is propor

tional to Q or both of them are functions only of sin or cos

Pro of From we nd

substituting it into we get

Solving it we have several options

Case I then so either Case

p

Case I I or Case I or

Case II then so either Case

p

II or Case I I or Case II Case I I coincides

with the Case I

Case III then from we get

Substituting it into the program we get

Solving the equation with resp ect to we get the following

cases

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 p ossible only for

For these values of and all p olynomials up to vanish and we

compute

After standard computation of resultants see B or Chapter for short

explanation of resultants technique we obtain that these three p olynomials

do not have common zero es

Lets consider the rest of all of these cases

then and In the Case I

P cos

Q sin cos cos

computing we get either or and in b oth cases we get

the comp osition with cos and sin resp ectively

In the Case I then

P cos cos

Q cos cos

and we get comp osition with cos

p p

case is In the Case I

similar

p p

P cos cos

Q sin cos sin cos

Performing computations we get and it is a comp osition with cos

In the Case I I then

P sin cos

Q sin cos

and we get comp osition with sin

p p

case is In the Case II

similar

p

P sin cos

Q sin cos sin cos

Performing computations we get and it is a comp osition with sin

In the Case I I I and then so

P sin cos cos

Q sin cos cos

Then from we get either and then P Q are comp ositions

with cos or then and Q P ie we again get

comp osition

Chapter

Center and Comp osition

Conditions for Ab el dierential

equations with p q algebraic

p olynomials of small degrees

Intro duction

In this chapter we summarize results on Ab el equation with p q algebraic

p olynomials which were obtained in BFY BY in the authors MSc

thesis and in the current PhD thesis All the results for which an explicit

reference is not given are new and belong to the current thesis so they are

given with pro ofs

We consider Ab el dierential equation

y pxy q xy

with px q x algebraic p olynomials in x Although this problem do es not

corresp ond directly to the classical center problem on a plane where p q are

trigonometric p olynomials it represents a signicantinterest by itself as it

helps to understand the combinatorial structure of the set of closed solutions

A center condition for this equation closely related to the classical center

condition for p olynomial vector elds on the plane is that for xed points

and a and for any solution y x y y y a ie all the solutions

return to the initial value on the interval a

As b efore we are lo oking for solutions of in the form

X

k

y x y y v x y

k

k

where y y y is the nite set of the co ecients of p

q Shortly we will write v x

k

The co ecients v are computed by formula and turn out to be

k

p olynomials b oth in x and We study the p olynomials ideals I R x

k

xv xg The problem is to nd conditions on p q I fv xv

k k

under which x a is a common zero of all I

k

Following BFY we intro duce p erio ds of the equation as those

C for which y y forany solution y xof The generalized

center conditions are conditions on p q under which given a a are

k

exactly all the p erio ds of

The ideal I C x is studied where is a set of co ecients of p oly

nomials p q

I fv xv xv xg I where I fv xv xv xg

k k k k

k

Our generalized center problem is the following

For a given set of dierent complex numbers a a a nd conditions

on algebraic polynomials p q under which these numbers ar ecommon zeroes

of I

Wesay that denes center on the set of numbers a a These

numbers are called p erio ds of since y y a for all the solutions

i

y xof

Comp osition conjecture for multiple ze

ro es and its status for small degrees of p

and q

The following comp osition conjecture has b een prop osed in BFY

and only if I has zero es a a a a if I

k k

k

Z Z

x x

P x ptdt P W x Qx q tdt QW x

k

Y

where W x x a W x is a p olynomial vanishing at a a a

i k

i

and P Q are some p olynomials without free terms W xis an ar

bitrary p olynomial

Suciency of this conjecture can be sho wn easily see BFY But we

still do not have any metho d to prove the necessity of this conjecture in

the general case although the connection between this conjecture and some

interesting 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 MSc thesis B and was com

pleted during PhD thesis

a The maximal numb er of dierent zero es of I ie the maximal number

of p erio ds of was estimated

Theorem Either the number of surviving dierent zeroes including

of I is less or e qual then deg P 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 deg P degQ

This result was proved in MSc thesis and is implicitly contained in com

putations given in BFY

The next results were obtained in MSc thesis B BY

b The ab ove stated comp osition conjecture was veried for the following

cases

deg P deg Q It was

done using computer symbolic calculations with some convenient represen

tation of P and Q Computations for the cases deg P deg Q

were p erformed together with Jonatan Gutman and Carla Scapinello work

ing on this summer pro ject under my sup ervision

Namelythe following theorem was proved

Theorem For the fol lowing cases the composition conjecture is true and

the fol lowing table gives the possible number of dierent periods in each case

deg P

deg Q

The pro of consists of computations of several rst x for each of the

n

cases considered equating x to zero and solving the resulting systems

n

of p olynomial equations on co ecients of p and q and on a It was done us

ing computer symb olic calculations using the sp ecial representation of P and

Q In most of the cases straightforward computations were far beyond the

limitations of the computer used Consequently some nonobvious analytic

simplications were used

For computations we used resultants technique Resultants provide us

with a con venienttoolforchecking whether n p olynomials of n variables

P x x C x x do not have common zero es

i n n

Consider one example Assume we are interested whether p olynomials

P x y Qx y R x y have common zero es

Claim Let Resultant P Q x S y Resultant R Q xS y If

Resultant S S y then P Q R do not have common zero es

there exists common zero x y of all p olynomials Pro of Assume

P Q RthenS y S y hence Resultant S S y Contradiction

The general construction for n p olynomials of n variables is exactly

the same

c On this base explicit center conditions for the equation on

were written in all the cases considered They turned out to be very simple

and transparent esp ecially in comparison with the equations provided by

vanishing of v See section for detailed description of the center

k

set

Description of a center set for p q of

small degrees

Consider again the p olynomial Ab el equation

y pxy q xy y y

with pxqx p olynomials in x of the degrees d d resp ectively We will

write

d

px x

d

d

q x x

d

d d

C

d d

Remind that v x are p olynomials in x with the co ecients p olynomially

k

d d

dep ending on the parameters C Let the center set C

d d

C consist of those for which y y for all the solutions

y x of

Clearly C is dened by an innite number of p olynomial equations in

v v In other words C is the set of zero es

k

C Y I of the ideal I fv v g in the ring of p olynomials

k

In this section we consider I as the ideal in C and not in C x

We x one endp oint a say a

The table of multiplicities obtained in B BY gives the number of

equation v necessary to dene C ie the stabilization moment

k

for the set of zero es of the ideals I x Since both v and are

k k k

p olynomials of degree k in the straightforward description of C

contains p olynomials of a rather high degree for example up to degree of

variables for the case deg P deg Q

The comp osition conjecture in contrast gives us very explicit and trans

parent equations describing this center set C Esp ecially explicit are equa

tions in a parametric form see below

The central set for the equation with deg p deg q has

been describ ed in BFY We remind this result here Let

px x x

q x x x

of the Abel Theorem BFY Theorem V The center set C C

equation is given by

The set C in C is determined by vanishing of the rst Taylor coecients

v v

Now let

px x x x

q x x

Theorem The center set C C of the Abel equation is given

by

The set C in C is determined by vanishing of the rst Taylor coecients

v v

Pro of By the comp osition conjecture which holds for this case p and

q belong to the center set if and only if P P W Q W where

xx P W W Thus W xx So we may assume Q

we get

Q xx x x

P xx xx x x x x

k

Comparing co ecients of x in b oth sides of equalities we get

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

If

px x

q x x x x

previous theorem one can prove the following then similarly to the

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 coecients

v v

Now let

px x x x x x

q x x

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

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

cients v v

Pro of By the comp osition conjecture which holds for this case p and

q belong to the center set if and only if P P W Q W where

W xx So we may assume Q xx P W W W

Thus we get

xx x x

x x x x x x xx xx xx

Hence

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

If

px 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 given

by

by vanishing of the rst Taylor coecients The set C in C is determined

v v

Now let

px x x x

q x x x x

Theorem BFY Theorem The central set C C of Abel

equation 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 Taylor

coecients v v

This theorem was proved in BFY using the fact that the comp osition

conjecture is true for this case The comp onent corre

sp onds to the prop ortionality of P and Q and the comp onent

corresp onds to the comp osition with W xx

Let

px 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 given

in a parametric form by

a

a

a a a

aa a a

a

or by

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

cients v v

Pro of We can represent P W Q W W where W xx

x a Thus

x x x x a x ax

x a x a x a x ax aa x

x x a ax x x

k

Comparing co ecients by x in b oth sides of equalities we obtain

After some transformations we obtain Notice that as

leading co ecient

Comp osition conjecture for some sp ecial

families of p olynomials

In this section we generalize MSc thesis results nding out some classes of

p olynomials p q of arbitrary high degree for which condition for a center on

agiven set of numb ers can b e explicitly found and coincide with comp osition

case

Let a a a b e given Consider any p olynomial W xvanishing

at all the p oints a j

j

Z

a

j

k

W dx and Theorem Assume that for at least one a

j

Z

a

j

n

k n

W dx Polynomials p W W q W W dene

center on a a if and only if

Z

a

j

k

W dx for at least one a Remark Notice that the condition

j

Y

is satised for instance for W x x a where all a are dierent

i j

i

Z

x

k

Indeed consider the function f x W t dt If all a j would

j

b e zero es of f x then deg f k But deg W sodegf x k

We obtain k k which is not satised for

Y

m

i

satises the condition Similarly one can showthatW x x a

i

i

Z

a

j

k

W dx for at least one a for almost all k and so on So this

j

condition is almost generic

Pro of Since x P x the conditions a imply

j

a imply Since x P x Qx the conditions

j

Theorem Assume that deg W and for at least one a

j

Z Z

a a

j j

n n

W W dx W dx

Z Z

det

a a

j j

nk nk

W W dx W dx

k n

Polynomials p W W q W W W dene center on

a a if and only if

Pro of The conditions a imply The conditions a

j j

imply

Z Z

a a

j j

n n

W W W

and the conditions a imply

j

Z Z

a a

j j

nk nk

W W W

If the determinant of the system is nonzero we get that the system has the

only zero solution

Chapter

Riemann Surface approach to

the Center problem for Ab el

equation

Intro duction

In this chapter Riemann Surface approach to the Center problem for Ab el

equation is discussed This is a convenient general setting where Ab el Equa

tion

dy dP z dQz

y y

dz dz dz

considered on agiven Riemann Surface In this chapter

we generalize notions of center and comp osition to the case of Riemann

surfaces

following Chapter we show that all the facts ab out Poincare return

map and recurrence relations to compute its co ecients remain valid

in this setting

Ab el equation on Riemann Surfaces

Let X b e a domain on a connected Riemann Surface and let P and Q be two

analytic functions on X For Ab el equation related to planar vector elds X

is a neighb orho o d of the unit circle S on C Laurent p olynomials P and Q

are analytic on X We consider the following Ab el dierential equation

on X

dy y dP y dQ A

A lo cal solution of A is an analytic function y on an op en set in X

such that the dierential forms dy and y dP y dQ coincidein

If x is a lo cal co ordinate in A takes the usual form

dy

y pxy q x

dx

where

d d

px P x Qx Qx

dx dx

Let Y X b e the universal covering of X The equation A can b e lifted

onto Y One can easily show that for any a Y and for any c C there

is a unique solution y of A on Y satisfying y a c whose singularities

c c

tend to innity as c tends to zero In what follows we always assume that

c is suciently small so y is regular and univaluedonany compact part of

c

Y but in general is m ultivalued on X

Denition Let be a closed curve in X We say that the Abel equa

tion A has a center along the curve if for any smal l c C y is

c

univalued along

Notice that in this denition it is sucient to assume that for any su

ciently small c and some a y a y a where y is a result of an

c c c

analytic contin uation of y along Indeed by uniqueness of a solution of the

c

rst order dierential equation A y a y a implies that y and y

c c c c

coincide in a neighb orho o d of a Then by analytic continuation y coincide

c

with y along the the whole ie y is univalued

c c

Denition We say that A has a total center on X if it has a

center along any closed curve in X

X simplyconnected Indeed In particular this is always the case for

in this case any closed curve is homotopic to a point so after analytic con

tinuation along any closed curvewe will obtain the initial value at this p oint

Denition 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 QxQw x We say that the Abel

equation A on X is induced from the Abel equation

Q A dy y dP y d

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 equivalentto

the words comp osition representation P xP w x QxQw x

When we deal with Ab el dierential equation we shall use the word factor

ization and when we deal with P and Q we shall use the word comp osi

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 ie y y w

Pro of Let y y takeavalue c at some p oint a X Consider the solution

c

y of A taking the value c at w a X By the invariance of the rst

c

dierential y w is a solution of A and it satises y w a c Hence

c c

lo cally y w y and analytic continuation completes the pro of

c c

Corollary If A is induced from A by w and if A has a center

along w then A has a center along

Denition Let a and b be two dierent points in X We say that

a and b are conjugate with resp ect to the equation A and with

to a certain homotopy class of curves joining a and b resp ect

in X if for any solution y of A with suciently smal l c y a c its

c c

continuation y along satises y b c In other words any solution

c c

of 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 p oints can app ear at all However the following prop osition gives a

basic reason for their app earance

Prop osition Let X X P Q P Q w be as above Consider two

dierent points a b X andapath joining them If w aw 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 two

aw b are conjugate along any joining them points a b with w

Pro of Follows immediately from the last lemma

Example Polynomial Comp osition Conjecture

in the case X C P and Q p olynomials in x

Here X is simplyconnected P and Q are analytic on X and hence the Ab el

equation

dy

px y q x y

dx

d d

with px P x q x Qx has a center along any closed curve

dx dx

As far as a factorization of is concerned assume that there exist

p olynomials w P Q such that P x P w x Qx Qw x Then

for X C is induced by w from

dy

P w y Q w y

dw

By prop osition any two points a and b such that w aw b are

conjugate 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 p oints

can app ear only in this way

The following conjecture was prop osed in BFY

Polynomial Comp osition Conjecture Two dierent points a and b in

C are conjugate for the Abel equation A with P Q polynomials in x if

and only if the fol lowing Polynomial Comp osition Condition is satised

there exists a factorization P x P w x Qx Qw x with polyno

mial mapping w such that w aw b

Notice that although the equation A is nonsymmetric with resp ect to

p and q this conjecture prop oses the symmetric condition if a and b are

conjugate for A then they are conjugate also for the equation

dy

q x y px y

dx

This situation is not unique in the center problem In Che it was shown

that the Lienard system

d y dy

g x f x

dx dx

R

x

has a center at the origin if and only if the functions F x f tdt

R

x

g tdt can be represented as a comp osition F x F z x Gx

GxGz x for an analytic function z x with z

Example Laurent Comp osition Condition in the



borhood of the unit circle S case X a neigh

fjxj g C P and Q Laurent series conver

gent on X

We may rewrite Ab el dierential equation in an invariant form

d dP dQ

i

then expressing sin and cos through x e ie

xx

cos

xx

sin

i

y

we obtain P and Q in the form of Laurent p olynomials in x and the

dierential 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 corresp onds directly to the classical CenterFocus Problem for ho

mogeneous p olynomial vector elds on the plane

The one of p ossible factorizations in this case whichwe shall call Laurent

Comp osition Condition takes the form P P w Q Qw where w

is a Laurent series and P Q are regular analytic functions on the disk D in

C containing the image w S

Lemma Laurent Composition Condition implies center along S

Pro of We have a factorization w X D and since D is simply con

nected the Ab el equation dy y dP y dQ on D has a total center then

by Corollary it implies center along S

In this case the conjecture that this is the only reason for center is not

true there are known cases of center along S for the equation A when

p and q can not be represented as a Laurent comp osition the example was

considered by Alwash in A see Chapter above

Nevertheless for pz q z Laurent p olynomials of small degrees up to

ie of the form z a z a z a z a z a the comp osition

representation is the only p ossible reason for center shown in Chapter

Poincare mapping

Here we discuss notions intro duced in Chapter in a more general setting

Let us return to a general situation of the Ab el equation A

dy y dP y dQ

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

and b

Let us consider a general solution y of the equation A along taking

c

the value c at the point a y a c For suciently small c we may

c

represent it in a form of a convergent power series in c

X

k

y xc v c

c k

k

where v v a x are multivalued functions on X x is a pointon

k k

For xed a substituting the solution y x in the form of expansion

c

P

k

y x c v x c into A we obtain the following recurrence

c k

k

relation on v x

k

v x

v x

and for n

v a and

n

X X

v xv xdQx v xv xv x dv xdP x

i j i j k n

ij n

ij k n

Denition The Poincare mapping C C of the equa

tion A along is dened by

cy b

c

where y is as above the solution of A taking the value c at the point a

c

and analytical ly continued along up to the point b

As it was shown

X

k

c c v b c

k

k

For X C a b we obtain the usual Poincare mapping as we

used in BY

Lemma The equation A has a center along unclosed curve with

end points a and b ie points a and b areconjugate if and only if v b

k

for any k Here v x for xed a are obtained by integration along

k

of

The equation A has a center along closed curve if and only if v x

k

are univalued functions along

Pro of Follows immediately from denitions of a center and Poincare map

ping

The recurrence relations can be in fact linearized in the following

sense Let us x the curve and the endp oints a and b and consider an

inverse function with resp ect to y and ctoy x

c

c y x y x y x y

c

One can easily see that for suciently small c and y we can represent y by

a convergent power series

X

k

xy c y x y y

k

k

where a since y a cc

k

Lemma BFY The coecients x satisfy the fol lowing recur

k

rence relation

x x and for k

d xk dP x x k dQx x

k k k

a

k

Pro of Write as

X

k

c xy

k

k

Taking a dierential of it with resp ect to x along the solution of A we get

X X X

k k k

d y k xy dy d xy

k k k

k k k

X

k

k dQx xy k dP x x

k k

k

Equating to zero the righthand side co ecients pro duces the required re

currence

Denition The Inverse Poincare mapping C C

of the equation A along is dened by

cy b

c

where y is as above the solution of A taking the value c at the point a

c

and c is chosen such that y bc Shortly id

c

shown As it was

X

cc bc

k

k

Notice that in our considerations we were free to cho ose the endp oint b

It leads immediately to the following

Lemma BFY For any q for any b C the ideals I

q

fv bv bg and I f b bg coincide

q q k

Pro of This follows immediately from the standard expressions for the

Taylor co ecients 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 ie points a and b are conjugate if and only

if b for any k Here x for xed a are obtained by

k k

integration along of the linear recurrence

Theorem The equation A has a center along a closed curve if and

e univalued functions along only if al l the functions x ar

k

Chapter

Factorization comp osition for

C the case X P and Q

rational functions

Intro duction

We started from the center problem for vector elds on the plane It was

reduced to the center problem for Ab el dierential equation with p q

Laurent p olynomials The natural question is to study Ab el equation with

p q rational functions Everything what can be said in general ab out

rational functions remains valid for the case of Laurent p olynomials

It turns out that the assumption that P and Q are rational functions

puts the comp osition factorization question completely in the framework

of algebraic geometry of rational curves and of the Ritt theory R We

show that the examples of Polynomial comp osition and Laurent comp osition

given in sections and are in some sense generic and any analytic

factorization of rational function can be reduced to comp osition of rational

functions

Comp osition and rational curves

The following facts are very basic in algebraic geometry We restate them

for convenience of our presentation For details we address the reader to any

classical algebraic geometry text eg Sha

Well denote by Y the curve in C parameterized by P Q

Y fP tQtt C g

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

Pro of We need to prove that for suciently large d there exists a p oly

nomial F x y of the degree d such that F P tQt Without loss

we may assume that P and Q have the same denominator of generality

i j d

R Consider all the pro ducts P t Qt R i j d These are p olyno

mials of t of the degrees less than or equal to i deg P j deg Q d deg R

dd

i j

ddeg P deg Q deg R But there exist such pro ducts P Q therefore

dd

for ddeg P deg Q deg R ie for ddeg P deg Q deg R

there exists a linear dep endence over C

X X

i j d i j

P t Qt R t hence P t Qt

ij ij

ij ij

P

i j

This p olynomial F x y x y vanishes on Y Such p olynomial of

ij

minimal degree denes an algebraic curve Y

ny subeld of a eld of rational functions is generated Luroth theorem A

by a rational function

Corollary There exist rational functions r t P Q st P t

P r t QtQ r t and st the map

C Y z P z Q z

denes a birational isomorphism between C and Y In particular Y is a

rational curve

Pro of

Notice that K C P tQt is is a subeld of the eld of rational

functions C t By Luroth theorem K C r t for some rational function

r t In particular P and Q b elong to K hence there exist rational functions

P t Qt such that P tP r t QtQ r t

It is obvious that is surjective and rational Lets prove that there exists

an inverse map Y C Since r t C P tQt there exist a ra

tional function R x y st R P tQt r t ie R P r t Qr t

r t Obviously it is a rational function from Y to C Let us prove that

R Indeed R id C C R z R P z Qz

R P r tQr t r t z since for any z there exists t z r t

Vice versa R id Y Y since R P tQt r t

P r t Qr t P tQt

Denition The degree of a map P Q C Y is the degree

of the algebraic extension C t C P Q

Denition The parameterization of a rational curve Y C Y

z P z Qz is cal led minimal if deg

From corollary it follows that a minimal parameterization denes a

birational isomorphism b etween C and Y

Denition The mapping f X Y is cal led not if there

exists an open set Y st each point of has more than one preimage

under f

Denition A rational function r is cal led common divisor under

comp osition of rational functions P and Q if P P r Q Qr The

common divisor r is cal led nontrivialif r is not

Denition A rational function r is cal led Comp osition Greatest

Common Divisor CGCD of rational functions P Q if r is a common

divisor under composition of P and Q and if r is another common divisor

of P and Q under composition then r R r for a rational function R

Denition The degree of a rational function is a maximum of

degrees of numerator and denominator

Lets notice that among rational functions only linear functions functions

of degree are Resp ectively if we are lo oking for nontrivial CGCD in

the class of rational functions it must have degree greater than It is easy

to show that CGCD exists and satises all the prop erties of usual greatest

common divisor In particular

Prop osition For any rational functions P t Qt their CGCD r t

exists and is given by corol lary CGCD is unique in the algebra of

rational functions under compositions up to composition with an invertible

rational function ie a function of degree ie two CGCD of a given

function can be obtained each one from another by a right or left composi

tion with a linear function

Pro of

Obviously r t from Corollary denes rational common divisor under

comp osition If r is another comp osition common divisor of P and Q then

P P r Q Qr so C r t C P tQt 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 rt 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 of

preimages 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 degree

of 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 priori

rational comp osition do es not add in fact anything new

Lemma deg if and only if there exists an analytic factorization

w C X where X is a Riemann Surface P and Q are analytic functions

on X such that P t P w t QtQw t and w is not

Pro of

Let deg then taking w r C C we get the required analytic fac

w C X torization Vice versa let there exist an analytic factorization

P t P w t Qt Qw t Then C P Q C P w Qw which

is a prop er subset in C t because w t glues some points in C but in

C t there are functions which map these p oints into dierent p oints Hence

C t C P Q

Corollary If there exists an analytic factorization of rational func

tions P P w Q Qw with w not then there exists a nontrivial

CGCD of P and Q P P r Q Qr for r of the degree greater than

Structure of comp osition in the case X

C P and Q p olynomials

We shall prove that the factorization in the form of Polynomial Comp o

sition Condition is essentially the only one natural factorization in the

p olynomial case namely

Theorem Assume there exists an analytic factorization P P w

Q Qw with w not Then there exists a polynomial factorization

P t P w t QtQw t with P Q w being polynomials the degree

of w is gre ater than and deg P Q

The pro of will followfromthe lemma

Lemma If we have a factorization P P r Q Q r with P t

Qt r t rational functions C C then there exists a linear rational

function C C st P Q r are polynomials

Pro of

tained essentially in R Let r a Let the degrees of This pro of is con

the rational functions r P be n m resp ectively The degree of the p olynomial

P will b e mn Then

r a with multiplicity not more than n P a with multiplicity

not more than m but P r with multiplicity exactly mn b ecause

P r is a p olynomial

Hence we get that r a with multiplicity n P a with multi

plicity m

Now take z ie a Then P with multi

z a

plicity n r with multiplicity m hence they are p olynomials

Similarly Q is a p olynomial

Pro of of theorem

By corollary there exists a factorization P tP r t QtQr t

for some rational functions P t Qt and rational function r t of degree

greater than such that deg P Q Then taking P P Q Q

w r we obtain the required p olynomial factorization P t P w t

QtQw 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 p olynomial case of Lienard system

Corollary If deg P Q s then the two polynomials P and

Q have a nontrivial CGCD of the degree s in the algebra of polynomials

under composition P t P r t QtQr twithP Q r algebraic

polynomials and deg r s The map z P z Q z denes a minimal

polynomial parameterization of the algebraic curve Y fP tQtt C g

Structure of comp osition in the case X

a neighborhood of the unit circle on C P

and Q Laurent p olynomials

We describ e p ossible factorization of Laurent p olynomials up to a natural

equivalence

Denition The composition representations P P r and P P r

are equivalent if there exists a linear r ational function st P P

r r

Theorem Up to the equivalence relation of denition there are

only two types of composition representations of a Laurent polynomial P

P P r where P is a usual algebraic polynomial and r is a Laurent

polynomial

k

P P r where P is a Laurent polynomial and r z for some k N

k

Any two composition representations of types and for deg r and

k are not equivalent

Pro of

P is a Laurent p olynomial hence has exactly two preimages and

Assume we are given a comp osition in the class of rational functions

P P r We shall show that bycho osing a suitable linear rational function

we obtain P P r where P P and r r are of

the 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 st a b

z b Then P P so P P

z

ab

is a Laurent p olynomial Then necessary r r

and there are no other p oints where r takes values and so r

k

r isan algebraic p olynomial of the form z for some natural k

If P has only one preimage of then similarly to the lemma

there exists a linear rational function st P P is a p olynomial Then

necessary r r and there are no other p oints

where r takes values and so r is a Laurent p olynomial

Since under comp osition with a linear function the number of preimages

ofagiven point can not change and are not equivalent

Corollary If deg P Q then P P r Q Qr with either

Laurent p olynomial comp osition P Q algebraic polynomials r

Laurent polynomial of degree greater than or

k

P Q Laurent polynomials r z for k

Chapter

Moments of P Q on S and

Center Conditions for Ab el

equation with rational p q

Intro duction

R R

i j i

The role of generalized moments of the form P Q dP and P qdz as

related to the centerFocus problem and comp osition conditions on the in

terval is investigated in BFY BY Recently a serious progress have

been done in this direction Pa N Chr

In this thesis we restrict ourselves to the case of Laurent p olynomials

and generalized momen ts on the circle

R R

It is shown that certain integral condition on P p and Q q

R

i j

namely vanishing of generalized moments P Q dP implies

jz j

center

Sucient center condition for Ab el equa

tion with analytic p q

We return to the case of Ab el equation A

dy y dP y dQ A

considered in a neighb orho o d of a unit circle S fjxj g in the complex

plane C with P Q analytic functions in some neighb orho o d of S not

necessary Laurent p olynomials

The following theorem is a summary of results due to J Wermer W

W The applicabilityof Wermers results to Center problem was discov

ered by JPFrancoise F

Theorem Wermer Let P Q be a pair of functions on the unit

circle S C Assume

P and Q are analytic in an annulus containing S and together separate

points on S

P on S

P takes only nitely many values more than once on S

R

i j

P Q dP for al l i j then there exists a Riemann Surface If

s

X and a homeomorphism S X such that S is a simple closed

curve on X bounding a compact region D such that functions P Q dened

on S by P P Q Q can be extended inside D to be analytic

there and continuous in D S

To use this theorem for our factorization we need to replace homeomor

phism by analytic map of a certain neighb orho o d of S into X

Lemma Let S C be a unit circle P and Q be analytic functions

in a neighborhood U of S such that P on S Let X be a Riemann

Surface P and Q regular functions on X and let S X be a

homeomorphism such that P P Q Q on S Then can be

extended as an analytic mapping of a certain neighborhood V U of S into

X with the same property P P Q Q in V

Pro of

P s hence P s so P y in a neighborhood of son X

We dene x in this neighb orho o d as y x P P x Lo cally

exists and is welldened Since these lo cal extensions agree on S they in

and dene a required extension on a certain neighborhood of S fact agree

Corollary If in the Abel equation A on C

dy y dP y dQ or dy y dQ y dP

P and Q are functions satisfying al l the properties and the do

main D provided by Wermers theorem is simplyconnected then the Abel

equation A has a center

Pro of

If P on S we may apply lemma Then the Ab el equation A is

induced by the analytic mapping from the Ab el Equation on X

dy y dP y dQ 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 has

a center along S

Notice that this condition is a sucient condition for center for Ab el

equation with arbitrary analytic co ecients But it is symmetric with resp ect

to P and Q although some of the center conditions for Ab el equation are

known to be nonsymmetric Below we shall explain it for the case of P Q

Laurent p olynomials

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 denes

the rational curve Y fP tQt t C g Image of a circle S under the

map is a closed curve on Y

Theorem Let P Q be rational functions without poles on S st at

of them has a pole least one of them has a pole inside S and at least one

outside S for instance Laurent polynomials Let S bound a compact

domain in Y Then deg

Pro of

Assume that deg Then consider a path in C joining two p oles of

P and Q inside and outside of S for simplicity and and intersecting

S only once at a regular point u We can assume also that do es

not contain preimages of double points in Y So for any x there are no

y 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 b ounded by

Then it S But it enters this domain since u is a regular point of

must intersect S at another p oint v uandwe get contradiction to the

choice of the path

Example P z z Qz

z

The rational curve Y is fxy g C and the curve S do es not b ound

a compact domain on it The degree of the map P Q is one and it is a

general situation for a map of degree one images of circles contracted to

p oles diverge on Y see gure with S fjz j k g S fjz j g

k k

k

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

k k

S−1 [P,Q]

S−2

G−1 G1 S 1 G−2 G2 S G3 S 2

3

Qz z Example P z z

z z

The rational curve Y is fx y g C the degree of the map is and the

curve S b ounds a compact domain on Y in fact S S

See gure for illustration images of the circles S cover Y twice b ecause

k

they have no space to diverge Arrows indicate directions of motion of

the curves S and G as k decreases from to

k k S -1

[ P , Q ] G--1 S -2 G 1

G2 G S -2 G S 3 1 3

S 2

Center conditions for the case of P Q

Laurent p olynomials

Theorem If P and Q are Laurent polynomials satisfying the condi

tion

Z

k n

P z Qz dP

jz j

for al l pairs kn of nonnegative integers and P Q together separate points

on S then P and Q can berepresented in the form of LaurentPolynomial

comp osition and hence the equation A has a center

Remark We b eliev e that for the case of Laurent p olynomials the Wermers

theorem remains valid without the assumption that P and Q together sep

arate points on S We hop e to present a pro of of this fact together with

a detailed investigation of Wermers surface for the case of P Q Laurent

p olynomials

Pro of

If b oth P and Q are algebraic p olynomials in z P and Q are represented as

a comp osition with z so we have a center

P and Q are Similarly if b oth P and Q are algebraic p olynomials in

z

represented as a comp osition with so we have a center Otherwise P and

z

Q have one p ole inside origin z and one p ole outside innity of S

Obviously P Q are analytic in a neighborhood of S and take only nitely

manyvalues more than once on S

Next we always can assume that P z for all jz j Indeed A

after the change of variables z u b ecame

dy

dP uy dQuy A

du

where P u P u Qu Qu both A and A have center

simultaneously But as P hasonlynitenumb er of zero es on C by rescaling

z z which do es not change a center for A we can assure that there are

no zero es of P on the circle S For example P z z has zero es of P

z

P z z but on S has not Under this change of variables the circle

z

jz j go es to juj but by Cauchy theorem for Laurent p olynomials

integrals 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 and

there exists functions P Q analytic inside D

Remind that wehave a map P Q C Y fP tQt t C g

Lemma If the curve S bounds a compact domain on X then the

curve S bounds a compact domain on a rational curve Y

Pro of

By lemma P Q is an analytic mapping dened in a neighborhood

of S which coincides there with P Q But hence P Q X C

maps a neighb orho o d of S into Y C By analytic continuation P Q

maps X into Y Hence P Q maps a compact domain D inside S onto

Y and the image of a compact domain under continuous mapping is com

pact Hence S is contained in a compact P QD Now one can easily

bounds a compact domain in Y show that in fact S

continue Pro of of theorem

By theorem deg P Q hence P and Q can be represented as a com

p osition P P w Q Qw If P and Q are usual algebraic p olynomials

we are done

k k

If not then P z P z Qz Qz for Laurent p olynomials P

Q But then on the rational curve Y we get P QS P QS so

P QS bounds a compact domain on Y fP tQt t C g

fP t Qt t C g Therefore by theorem deg P Q so they are

represented as a comp osition

If we again obtain their representation as a comp osition of Laurent p oly

n

nomials with z we rep eat our considerations and nally we are left with

the comp osition P P w Q Q w with w Laurent p olynomial P

N

and Q algebraic p olynomials It gives us the comp osition P P w z

N

Q Q w z and we are done

Chapter

Generalized Moments and

Center Conditions for Ab el

dierential equation with

elliptic p q

Intro duction

In Chapter we considered Generalized moments for rational functions P

and Q and have shown that vanishing of all generalized moments implies

comp osition representabilityofP and Q and hence center for Ab el equation

A

dy y dP y dQ A

In the pro of we essentially used the fact that P QC is a rational curve

ie algebraic curve of genus

Suggested generalization is to consider the same question for Laurent

series instead of Laurent p olynomials The rst nontrivial case of Laurent

series in our algebraic context is the case of elliptic functions For P Q

R

i j

elliptic functions weshow that all the generalized moments P Q dP along a

small curve around the origin vanish But at the same time we demonstrate

that in general Ab el equation A with P Q elliptic functions do es not

have 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

do es not hold

It can be explained by the fact that elliptic curve X P QC is an

algebraic curve of genus and hence it is homologically nontrivial S is

emb edded by the map P Q into this curve in a homologically trivial way

and hence all the moments on S vanish But the domain b ounded by

P QS is not simplyconnected Hence multiple integrals along this curve

do not vanish and A do es 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 QC is a torus with small circle

around the origin dividing it into two parts compact topological ly nontrivial

part on which and are regular have no p oles and the second part wich

is top ologically equivalent to C without origin but and have a pole at

zero

picture

Generalized Moments on Elliptic Curves

As it was shown ab ove Chapter the study of the center problem for the

Ab el equation leads naturally to the momentlike expressions of the form

R R

i i

P dQ and Q dP and similar more complicated ones

All these expressions naturally app ear as a part of the expressions for the

Taylor co ecients of the Poincare mapping given in Chapter ab ove

Transforming these co ecients via integration by parts one obtains ex

R R

i j i j

pressions of the form P Q dP and P Q dQ and the expressions

Z Z Z

x x x

n

i i i

n

P x q x dx dx m x P x q x P x q x

n n n i i i

n

where pxdP xdx q xdQxdx intro duced by JP Francoise

Expressions of this form considered on complex curves of genus greater

than zero lead to multivalued functions Hence we generalize denition of

moments to this case

Denition For the two multivalued functions P Q on a Riemann

Surface X we dene generalized moments m and m as integrals along

ij

ij

a given curve of

i j

dm P Q dQ i j

ij

i j

dm P Q dP i j

ij

We normalize m and m assuming that al l of them vanish at a xedpoint

ij

ij

a Here P and Q are analytical ly continued along the curve from the point

a

Remark Let us notice that

i j i j i i

P Q P Q P P j

j j

dm Q d dQ d dm d

ij

ij

i i i i i

Below we shall use this relation in computations of m

ij

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 al l the

moments m x for a certain set of indices i j are univalued functions

kn

along a given curve and if is not closed with end points a and b then a

and b are conjugate ie m am b under analytic continuation along

ij ij

Belowwe present some rather preliminary computations in this direction

for elliptic functions

Theorem Let P Q be a pair of el liptic functions with the only pole

at zero Then for al l nonnegative k n and al l suciently smal l inside the

paral lelogramm of periods the generalized moments

Z

k n

P u Qu dP

juj

vanish

Pro of The pro of follows from the fact that for suciently small smaller

than min elliptic functions P and Q do not have poles in the par

allelogramm of p erio ds outside the circle juj Hence the circle juj

b ounds on the elliptic curve a compact domain on which the integrand is

regular

Remark We may show this fact using combinatorial arguments Remind

some basic notions from the theory of Weierstrass function In lo cal co or

dinates z around zero on C it can be dened by the formula

X

i

z c z

i

z

i

where co ecients c are computed by recursion

i

P

i

c c

j ij

j

c

i

i i

Here the rst co ecients are dened as c g c g where g

and g dep end only on p erio ds

X X

g g

where summation is taken over all nonzero p eio ds of

The following combinatorial formulas hold

g

g g

Now the pro of of the theorem follows from the fact that any in

tegral of the form can be presented as a sum of integrals of the form

R R

k k

u u du and udu Both integrals are zeros the rst be

juj juj

ing the incrementof along juj while the second contains only even

k

degrees of z and hence do es not have residue inside juj

Let us notice that from remark follows that generalized moments

R R R

j i

Q P

k n k n

P u Qu dP and P u Qu dQ are in equal as d

i juj juj

Next question weareinterested in is computation of generalized moments

along p erio ds of elliptic integrals

Theorem Let be Weierstrass function be periods Then

Z

n k

u u du

i

for al l k and n

R R R

n k n k

Pro of du g g du F du

i i i

WILL BE ADDED

R

n k

Computations for u u It is known that in general generalized

i

moments along p erio d is equal to

Z

A B

i i

i

R

udu well nd these A and B where

i

i

Ab el equation with co ecientselliptic func

tions

R

k n

As wesaw in the previous section all the generalized moments P u Qu dP

juj

pole at the origin vanish For ra for elliptic functions P Q with the only

tional functions that would imply center under some additional restrictions

on P and Q For elliptic functions the similar statement do es not hold Al

though for some elliptic P Q the equation A mayhavecenter say P Q

in general A do es not have a center In particular the following theorem

holds

Theorem Let be Weierstrass function The equations

dy y d y d and

dy y d y d

ramify along any smal l circle around zero

Pro of By theorem Chapter in order for Ab el dierential equa

tion A to have center all the functions z in the expansion of Poincare

k

Return Map must b e univalued functions of z Substituting nite expansion

for suciently many c to garantee all the necessary terms for all negative

i

z we nd that for powers of z for z into formulas of Chapter for

k

Ab el equation z has logarithmic term g g lnz which is

the determinant of the Weierstrass function and hence nonzero ie z

is nonunivalued function

For Ab el 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 lnz and it is nonzero

WILL BE ADDED

Remark Well add handwriting pro of of this fact Also explanation

why and have dierent vanishing b ecause of

k

Remark Another remark is that in contrast with the notions of center

and conjugate points for the Ab el equation on a plane where all the deni

tions dep end on homotopy class of the path here we meet with homology

i j i j

e integrate the closed forms P Q dP P Q dQ several class of Here w

times

REFERENCES

A MAM Alwash On a condition for a centre of cubic nonautonomous

equations Pro ceedings of the Royal So ciety of Edinburgh A

AL MAM Alwash NG Lloyd Nonautonomous equations related to

polynomial twodimensional systems Pro ceedings of the Royal So cietyofEd

inburgh A

B M Blinov Some computations around the center problem related to

the algebra of univariate polynomials MSc Thesis Weizmann Institute of

Science

BFY M Briskin JP Francoise Y Yomdin The Bautin ideal of the

Abel equation Nonlinearity No

BFY M Briskin JP Francoise Y Yomdin Center conditions com

positions of polynomials and moments on algebraic curves Ergo dic Theory

Dynam Systems no

BFY M Briskin JP Francoise Y Yomdin Center conditions II

Parametric and model center problems Israel J Math

BFY M Briskin JP Francoise Y Yomdin Center conditions III

Parametric and model center problems Israel J Math

BFY M Briskin JP Francoise Y Yomdin Generalized Moments

Center Focus Conditions and Composition of Polynomials Op erator the

ory system theory and related topics BeerShevaRehovot

Op er Theory Adv Appl Birkhuser Basel

BY M Blinov Y Yomdin Generalized center conditions and multi

plicities for the polynomial Abel equations of smal l degrees Nonlinearity

BY M Blinov Y Yomdin Center and Composition Conditions for

Abel Dierential Equation and rational curves Qualitative Theory of Dy

namical Systems

BY M Blinov Y Yomdin CenterFocus and Composition Condition

for Abel Dierential Equations on Riemann Surfaces and associated Gener

alized Moments in preparation

Che L Cherkas Number of limit cycles of an autonomous secondorder

system Dierentsialnye uravneniya No

Che L Cherkas On the conditions for a center for certain e quations

of the form yy P xQxy R xy Dierential equations

Chr C Christopher An algebraic approach to the Classication of Cen

ters in Polynomial Lienard Systems J Math Ann Appl

Chr C Christopher Abel equations composition conjectures and the

model problem Bull Lond Math So c No

Dev JDevlin Word problems related to periodic solutions of a nonau

tonomous system Math Pro c Camb Phi So c

Dev JDevlin Wordproblems related to derivatives of the displacement

mapMath Pro c Camb Phi So c

ER A eremenko LRub el The arithmetic of intire functions under com

position Advances in Math

F JPFrancoise Private communications

GL A Gasull J Llibre Limit cycles for a class of Abel equations SIAM

JMath Anal No

dx

L A Lins Neto On the number of solutions of the equation

dt

n

X

j

a tx t for which x x Inventiones math

j

j

N N Roytvarf Generalized moments composition of polynomials and

Bernstein classes in Entire functions in mo dern analysis BYa Levin

memorial volume Isr Math Conf Pro c

Pa F Pakovich A counterexample to the Composition Conjecture

to app ear in Pro c AMS

Pa F Pako vich On the polynomial moment problem preprint

R J Ritt Prime and composite polynomials Trans AMS

Sha I Shafarevich Basic algebraic geometry SpringerVerlag

Sch D Schlomiuk Algebraic particular integrals integrability and the

problem of the center Trans AMS No

W J Wermer Function Rings and Riemann Surfaces Annals of Math

n

W J Wermer The hul l of a curve in C Annals of Math