The Classification of the First Order Ordinary Differential Equations with the Painlevé Property

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The Classification of the First Order Ordinary Differential Equations with the Painlev´e Property

The Classical and a Modern Algebro-Geometric Approach

Version 1.0 Georg Muntingh

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Preface and Acknowledgments

I would like to acknowledge my considerable debt to many people who helped me.

First of all I want to thank my supervisors, Marius van der Put and Jaap Top. Their doors were always open, and they never seemed to tire of ex- plaining something to me for the second or even the third time, encouraging me to keep trying. Secondly my gratitude goes out to Professor Masahiko Saito from Kobe University for his talk in Utrecht that indirectly led to this thesis and his help with references for first order equations.

Several people are responsible for making the text more readable and clear, both on the mathematical and on the linguistical part. For that my thanks go to Monique van Beek, Jeroen Sijsling, Laurens van der Starre and of course my girlfriend Annett. Moving to Oslo to live with her formed the primary motivation to finish up my thesis. Furthermore I am of course indebted to my family for supporting me all my life.

I would like to thank several people who provided me with the facilities I needed to write this thesis, in particular the system operators Harm Paas, Jurjen Bokma and Peter Arendz for the GNU/Linux Debian system at the Department of Mathematics at the Rijksuniversiteit Groningen, the clean- ing lady Anja who was always cheerful in the morning and Ineke from the administration for the invigorating chats and the many cups of coffee.

For typesetting I used L^ATEX 2ε and the very convenient L^ATEX editor Kile. The pictures were made with The Gimp, Gnuplot and Dia, and the frontispiece was inspired by the logo of Wikipedia.

The first chapter will serve as an introduction to Painlev´e Theory, giving some motivation and intuitive definitions. At the end of the chapter several questions will be posed that will be discussed later in the thesis. The second chapter will deal with a large part of the mathematics that is needed later on, especially in the chapter on modern theory. In the third chapter an

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overview of the historical development of Painlev´e Theory will be given, together with rewritten classical theorems and a rewritten classical proof.

After that, in the fourth chapter, a detailed modern theory will be presented, followed by the fifth and final chapter containing conclusions and suggestions for future work. At the end of the document one can find an index of some terminology and names, referring to the page where they occurred first, and a list of symbols accompanied by a short description.

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LIST OF FIGURES

1 A portrait of Paul Painlev´e from 1929 . . . 1 1.1 A picture of the Riemann surface corresponding to the loga-

rithm, above a disk in the complex plane. . . 3 2.1 Several structures of functions together with their relations. . 11 3.1 Two 19th century mathematicians who layed the foundations

for Painlev´e Theory. . . 20 3.2 Two mathematicians who solved the general case for first or-

der equations. . . 22 3.3 Two mathematicians who redid some classical Painlev´e Theory. 24 4.1 A schematic representation of the situation in the proof of

Proposition 4.28. . . 50 B.1 A caricature of Paul Painlev´e from 1932 . . . 69

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Figure 1: A portrait of Paul Painlev´e from 1929

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CHAPTER 1 Introduction

Les Mathématiques constituent un continent solidement agencé, dont tous les pays sont bien reliés les uns aux autres; l’oeuvre de Paul Painlevé est une ˆıle originale et splendide dans l’océan voisin,– H. Poincaré

T

he class of all differential equations is enormous and very complicated to study in general. The best one can do is to restrict our research to a class of differential equations that is easy enough to say sensible things about and wide enough to describe a wide spectrum of phenomena. This thesis is about such a class, namely the class of first order ordinary differential equations with the Painlev´e Property.

Before we define the Painlev´e Property, we give in Section 1.1 some conceptual definitions of the problematic points of a differential equation.

Armed with these notions, we give a conceptual definition of the Painlev´e Property in Section 1.2 and motivate why we should study equations with this property. Finally, in Section 1.3, we shall restrict ourselves to the case of first order ordinary differential equations and discuss what kind of questions we can ask ourselves. The remainder of the thesis will then be concerned with the answers to these questions.

1.1 Problematic Points

In this section, we will look at some of the problems that can occur regarding the solutions of differential equations. To be more precise, we shall introduce three types of so-called problematic points. When we know which problems can occur in this context, we can restrict ourselves to differential equations that do not have these problems and try to examine this simplified case.

This is exactly what is done in Painlev´e Theory.

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Figure 1.1: A picture of the Riemann surface corresponding to the logarithm, above a disk in the complex plane.

An example of a problematic point is a so called branch point . This is a point in which multivaluedness of the solution occurs. This means that there does not exist a neighborhood of the point in the complex plane in which we can define a solution, but there does exist a neighborhood in the universal covering space on which we can. Let us illustrate this with an example.

Example 1.1 (Branch point of the logarithm). Consider the initial value problem

f⁰(z) = 1

z, f (1) = 0.

In a neighborhood of the point 1, this equation clearly has a solution f (z) :=

log z. From function theory, however, we know that we cannot get a power series solution in a neighborhood of 0. Furthermore, we cannot extend the solution in a neighborhood of 1 to a function that has a power series expansion in a punctured neighborhood of 0. So what can we do? The answer is that we can construct a larger space, called a Riemann surface, on which a solution can be defined. Maximal solutions that we were able to define on the complex plane, for instance the analytic function log : C\R≤0 → C that restricts to the ordinary logarithm on R>0, then appear as projections to C of restrictions to a certain sheet of our grand solution on the Riemann surface (see figure 1.1).

In general we call a point a branch point of the differential equation if it has a punctured neighborhood U in which a power series solution of the differential equation can everywhere determined locally, yet there exists no

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global solution on U restricting to a power series everywhere, no matter how small U is chosen.

Another problematic point is a pole. This is a point for which the solution goes to infinity when we approach this point, but there exists a monomial z^k such that the product of z^k and the solution can locally be expressed as a convergent power series. Another way to say this is that the function has a pole in 0 if it can locally be expressed as a Laurent series in z. An easy example of a differential equation with a pole is the following.

Example 1.2 (Pole). Consider the initial value problem f⁰(z) = −1

z², f (1) = 1.

This initial value problem clearly has a unique solution f (z) := ¹_z, which goes to infinity at z = 0. If we multiply this solution by z¹, then we obtain the function that equals 1 in every point. This function is clearly the convergent power series 1 + 0 · z + 0 · z²+ · · · . Therefore this initial value problem has a pole at z = 0.

Another type of singularities is the type of essential singularities. These are much worse than poles.

Example 1.3 (Essential singularity). Consider the initial value problem f⁰(z) = −1

z²f (z), f (1) = e.

This equation has a unique solution f (z) := e^1/z, which has a singularity at z = 0. Since there does not exist a natural number k such that z^ke^1/z is a convergent power series at z = 0, this is an essential singularity.

We shall call all these three types of points problematic points. The last two, where the function becomes infinity, are called singularities.

1.2 The Painlev´ e Property

In the previous subsection we discussed what kind of problematic points can arise in solutions of differential equations. In this subsection these points will be the ingredients of our definition of the Painlev´e Property.

We say that a problematic point of an initial value problem is movable, if it changes position when we (slightly) change the initial condition (that is, if we change the f₀ in the initial condition f (z₀) = f₀, and no matter how small this change is). If a problematic point is not movable, then we say it is fixed .

Definition 1.4 (Painlev´e Property). A differential equation has the Painlev´e Property (PP) if it has no movable branch points and no movable essential singularities.

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Remark 1.5 (Why poles are not that bad). Note that fixed problematic points and even movable poles are allowed for a differential equation with the PP.

The reason for this is that the natural object for differential equations to live upon is not the complex plane, but the Riemann sphere P. This is the complex plane together with an additional point at infinity, and it can be drawn as a sphere whose top represents the point infinity and whose bottom represents the point zero. If we are interested in local properties, as we are with local solutions of the differential equation, then we only have to look at a piece of the Riemann sphere (a chart ). Switching between the top chart and the bottom chart then comes down to substituting z⁻¹ for z. So if a solutions has a pole at zero, then we can write it locally as a Laurent series, and if we switch to the chart at infinity we obtain a nice power series.

Example 1.6. The problematic points of the Examples 1.1, 1.2 and 1.3 are all independent of the value of the initial condition. Therefore the differential equations from these examples have the PP.

where f⁽ⁿ⁾(x) denotes the functions resulting after taking the derivative n times

Remark 1.7. Linear differential equations are equations of the form a0(x)f (x) + a1(x)f⁰(x) + · · · + an(x)f⁽ⁿ⁾(x) = b(x),

where f⁽ⁿ⁾(x) denotes the function resulting after taking the derivative of f (x) n times. The equations of the Examples 1.1, 1.2 and 1.3 are all linear, and they have the PP. The set of linear differential equations has been studied extensively, and one can show that they all have the PP.

Remark 1.8. Though it is true that many equations in physics can be approx- imated by linear equations, most of them are highly nonlinear. Therefore the need arises to study a class of equations that captures a wide spectrum of these nonlinear phenomena but is still mathematically easy enough to say sensible things about. The equations satisfying the PP seem to be highly appropriate for this. In physics, there are a lot of models given by an equation with the PP. For a list of areas in physics in which the PP occurs, see Appendix B. This huge list suggests that the PP is a frequently occurring notion in physics, indicating that it has some physical meaning. In this sense, the equations with the PP form a very important class to study.

Example 1.9. As an example, we ask ourselves for which positive integers p and q the equation

f^0p = f^q, gcd(p, q) = 1 (1.1)

has the Painlev´e Property.

We can restrict to initial conditions in the point 0, because the equation is autonomous. That is, the equation depends only on f⁰ and f and not on

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z. Suppose that we have a solution w of this equation for an initial condition w(0) = w0 6= 0. Thenin a neighborhood of 0 w satisfies one of the initial value problems

w⁰ = ζ_p^kw^q/p, w(0) = w0 6= 0, k = 0, . . . , p − 1, (1.2) where ζ_p satisfies the equation ζp^p = 1. Therefore we can find all local solutions of (1.1) by finding all solutions of (1.2).

Now what do these solutions look like? Let us try to find some solutions of (1.2). Separating variables and integrating gives us

z(w) = Z

ζ_p^−kw^−q/pdw + const = ζ_p^−k p

p − qw^p−q^p + const.

Inverting this function, we find solutions w(z) =

ζ_p^kp − q

p (z − const)

_p−q^p .

Combining this with the initial condition w(0) = w06= 0, we find a solution w_k(z) =

ζ_p^kp − q p z + w

p−q p

0

_p−q^p

to this initial value problem, which is analytic in a sufficiently small neighborhood of 0. For certain values of p and q it has a branch point at z = −ζ_p^−k_p−q^p w

p−q p

0 , depending on the initial condition w(0) = w₀. Using that gcd(p, p − q) = 1, we find that the criterion for the absence of (movable) branch points becomes:

f^0p = f^q has no movable branch points ⇐⇒ |p − q| = 1.

What about the initial condition w(0) = 0? This is the remaining case for which the differential equation might have solutions. Such a solution is either the zero function (which has of course no branch points or essential singularities) or a function that is in some point unequal to 0. However, then it is a solution to an initial value problem with w(0) 6= 0 as well, in which case it has no movable branch points or essential singularities.

1.3 First Order Equations with the PP

Having defined the PP for any differential equation in the previous section, we shall from now on restrict ourselves to first order ordinary differential equations. More precisely, we shall in the remainder of the thesis consider differential equations of the form

F (z, y⁰, y) = 0,

with F a rational function of y⁰ and y and some algebraic function of z. We can then formulate the following questions.

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(a) The mathematician Ja- copo Francesco Riccati from the Venetian Republic (1676- 1754).

(b) The German mathematician Karl Theodor Wilhelm Weierstrass (1815-1897).

1. How can we determine if such an equation has the PP?

2. Can we give a precise classification of all equations of this form with the PP?

3. If we can give such a classification, how can we, given a differential equation, effectively determine which of the categories of the classification it belongs to?

4. If we can give such a classification, can we, given a differential equation with the PP, effectively determine what kind of transformation leads to the canonical form in its category?

5. Can we give a modern equivalent of the PP?

It is this type of questions that this thesis tries to answer. The answer to the first question is known in the classical literature as Fuchs’s Criterion, and we shall discuss it in Sections 3.1 and 3.4. After we give a modern description of first order Painlev´e Theory in Chapter 4, we can also obtain an answer from standard algorithms in algebraic geometry.

Question (2) was completely solved in a classical way at the end of the 19th century [17], and in Chapter 4 we answer the question in a more modern way using some differential algebraic geometry. It turns out that by means of appropriate transformations we can transform such an equation with the PP into either a Riccati equation

y⁰ = a(z)y²+ b(z)y + c(z),

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or a generalized Weierstrass equation, y⁰²= a(z)(4y³− g₂y − g₃).

But this classification directly gives rise to question (3). Again after having obtained a modern description, it becomes clear that the differential equations are classified by the genus of a certain associated curve, and then standard algorithms from geometry give the answer.

The answer to question (4) was written down very precisely by Malmquist [14] but probably also already by Painlev´e [17]. The modern answer to this question will be that each differential equation reduces to one of the categories by means of a transformation that induces an isomorphism on the associated differential function field . This will be made somewhat more clear in a remark at the end of Section 4.1.

The final question was already answered by Matsuda [15] but without any motivation. In the Sections 4.5 and 4.6 it will be derived that an equation has the PP if and only if the “derivation on the associated function field is regular .” What we precisely mean by this will become clear in the last section of the next chapter.

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CHAPTER 2 Prerequisites

I

n this chapter we shall discuss the prerequisites for the remainder of the thesis not taught in standard undergraduate courses. We assume that the reader is familiar with some notions encountered in a first course in commutative algebra, algebraic geometry and function theory.

We start in the first section with looking at several structures of functions that we shall frequently use throughout the thesis. In the next section these notions will be applied to find an explicit description for the factorization of a polynomial differential equation. This will be used in the next chapter. After this, two short sections on local theory follow, at the end of which we are able to give a precise definition of ramification (and thereby branch points). The final section is about function fields with a derivation on it. Such differential function fields will replace the notion of a differential equation in Chapter Modern Painlev´e Theory.

2.1 Fields of Functions

Throughout the thesis we shall consider several types of functions, such as polynomials, power series etc. Each of these types of functions forms a class that carries some algebraic structure, and these algebraic structures themselves are related to each other by constructions from algebra. In this section we shall describe the functions we need, describe the algebraic structures they belong to and describe the connection between these algebraic structures.

Let R be a ring and k be a field. We start by giving examples of rings and fields whose elements can be interpreted as (or more precisely induce) functions. In the rest of the thesis R and k will often denote the complex numbers, so we could take R = k = C in the example below. Moreover

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every ring encountered in this thesis has characteristic zero.

Example 2.1. (1) A polynomial in the variable X with coefficients in R is an expression of the form

a₀+ a₁X + · · · + a_nXⁿ, n ∈ N0, a_i∈ R,

and all such polynomials together form the ring R[X] of polynomials with coefficients in R.

(2) A rational function in the variable X with coefficients in R is an expression of the form

a₀+ a₁X + · · · + a_nXⁿ

b₀+ b₁X + · · · + b_mX^m, n ∈ N0, a_i, b_i ∈ R,

and all such rational functions together form the field R(X) of rational functions with coefficients in R.

(3) An algebraic function in the variable X over k[X] is a function f (X) defined on some domain of k that satisfies a polynomial equation

0 = P₀(X) + P₁(X)f (X) + · · · + P_n(X)f (X)ⁿ, P_i(X) ∈ k[X], and all such algebraic functions together form the field k(X) of algebraic functions over k[X].

(4) A formal power series in the variable X with coefficients in R is an expression of the form

∞

X

k=0

a_kX^k= a₀+ a₁X + a₂X²+ · · · , a_i ∈ R,

and all such formal power series together form the ring R[[X]] of formal power series with coefficients in R. (Multiplication is defined byP

ka_kX^k· P

kb_kX^k =P

k

Pk

n=0a_nb_k−n

X^k, as in the case of polynomials.) A formal power series is called a convergent power series if it has a positive radius of convergence r := lim_n→∞inf |a_n|^−1/n. The convergent power series together form the ring R{X} of convergent power series.

(5) A formal Laurent series in the variable X with coefficients in k is an expression of the form

∞

X

k=n

a_kX^k= a_nXⁿ+ a_n+1Xⁿ⁺¹+ · · · , a_i ∈ k, n ∈ Z,

and all such formal Laurent series together form the field k((X)) of formal Laurent series with coefficients in k. A formal Laurent series is called a convergent Laurent series if it has a positive radius of convergence. The convergent Laurent series together form the field k({X}) of convergent Laurent series.

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k[[X]]

formal power series k{X}

convergent power series k[X]

polynomials

k((X)) formal Laurent series

k({X})

convergent Laurent series k(X)

rational functions

lim−→k((X^1/d)) formal Puiseux series

lim−→k({X^1/d}) convergent Puiseux series

k(X) algebraic functions

...take quotient field . ......................

...take algebraic closure. ......................

...

. .......

......

...

.. ...

...

......

...

. .......

......

...

.. ......

......

...

. .......

......

...

.. ......

......

...... .................... .

. . .. . .. ........

...... .................... .

. . .. . .. ........

........ .................. .

. . . . .. ... ......

........ .................. .

. . . . .. ...

......

...... .................... .

. .. . . .. .....

...

...... .................... .

. .. . . .. .....

...

Figure 2.1: Several structures of functions together with their relations.

(6) A formal Puiseux series in the variable X with coefficients in k is an expression of the form

∞

X

k=n

a_kX^k/d= a_nX^n/d+ a_n+1X^(n+1)/d+ · · · , a_i ∈ k, d ∈ N, n ∈ Z,

and all such formal Puiseux series together form the field of formal Puiseux series with coefficients in k, which we shall denote by k((X)) or lim−→k((X^1/d)) in fancy algebraic notation. A formal Puiseux series is called a convergent Puiseux series if it has a positive radius of convergence. The convergent Puiseux series together form the field of convergent Puiseux series, which we shall denote by k({X}) or lim−→k({X^1/d}).

(7) An analytic function, or holomorphic function, is a function from some domain U ⊂ C to C that equals in some neighborhood of each point z0 ∈ U some convergent power series P

kak(z − z0)^k. For a fixed domain U ⊂ C, all such analytic functions U → C together form the ring of analytic functions on U .

(8) A meromorphic function is a function f from some domain U ⊂ C to C t {∞} for which there exists a discrete set of points V ⊂ U such that fU \V

is an analytic function. For a fixed domain U ⊂ C, all such meromorphic functions U → C t {∞} together form the field M(U ) of meromorphic functions on U .

Remark 2.2 (Generalization to more variables). If we take the polynomials in the variable X₂ with coefficients in the ring R[X₁], we get the polynomials in the variables X₁ and X₂with coefficients in the ring R. In this way we can inductively define the ring of polynomials in the variables X1, . . . , Xn with coefficients in the ring R, which we shall denote by R[X₁, . . . , X_n]. In the

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same way we can generalize all other classes of functions from the previous example to a finite number of variables.

Remark 2.3 (Identification of polynomials and functions). Strictly speaking, the polynomials cannot be identified with functions, because for some rings R several polynomials can induce the same function (the same holds for other types from Example 2.1). An element of R[X] induces a function R → R, and some of these functions are represented by elements of R[X]. For the infinite rings R we shall use there will always be a unique representative.

Therefore we use the term polynomial both for an element of R[X] and its induced mapping R → R.

Remark 2.4 (Shifting the variable). Sometimes we want to identify for instance k{X} with the convergent power series functions in a point a ∈ k.

To make this identification then more explicit, we shall write k{X − a} for k{X}. Changing the symbol does not change the mathematics, so this is just some convenient notation.

The following theorem gives us the relations between the several structures of functions from Example 2.1, and similar results hold when there are more variables. For a schematic representation of these relations, see Figure 2.1.

Theorem 2.5. We have the following relations between the structures of functions from Example 2.1.

1. The field of fractions of k[X] is k(X).

2. The field of fractions of k{X} is k({X}).

3. The field of fractions of k[[X]] is k((X)).

4. The algebraic closure of k(X) is k(X).

5. The algebraic closure of k({X}) is lim−→k({X^1/d}).

6. The algebraic closure of k((X)) is lim−→k((X^1/d)).

Proof. The first four statements are by definition or immediate. For a proof of the statements (5) and (6), which constitutes the Puiseux Theorem, see [25, Theorem 3.1] for an elementary and constructive proof. Van der Waer- den gives a shorter proof in [24, II, par. 14] that is due to Ostrowski.

2.2 Factorizing a Polynomial Differential Equation

In the previous section we discussed several structures of functions and the connection between these structures. In this section we shall use this to find the factorization of a polynomial differential equation. We shall use the following immediate consequence of the Puiseux Theorem:

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Corollary 2.6 (Factorization of a polynomial equation). Suppose we have a polynomial

F (S, T ) = P₀(T )S^m+ · · · + P_m(T ), P₀(T ) 6≡ 0, F ∈ C[S, T ].

For a fixed a ∈ C, there exists a unique set {S1(T ), . . . , S_m(T )} ⊂ C(((T − a)^1/d)) such that

F (S, T ) = P0(T )

m

Y

i=1

(S − Si(T )) .

The form of a Puiseux series S_i(T ) from the previous corollary depends on P0(T ) and on the S-discriminant ∆(T ), which are both polynomials in T . We shall need the following theorem in Chapter Classical Painlev´e Theory.

Theorem 2.7. In the setting of the previous corollary, with F ∈ C[S, T ] irreducible, we have that the Si(T ) ∈ C({(T − a)^1/d}). Furthermore, we have the following.

• If P₀(a) 6= 0 and ∆(a) 6= 0, then the S_i(T ) have no terms with negative exponents and no terms with nonintegral exponents. That is, the S_i are power series in T .

• If P₀(a) 6= 0 and ∆(a) = 0, then the S_i(T ) have no terms with negative exponents. That is, the Si(T ) are power series in T^1/d for some d.

• If P₀(a) = 0, then there exists a Si(T ) starting at a term with negative exponent.

Proof. For every but the last claim, see for instance [24, Sections II.13- 14]. For the last claim, suppose that all Si(T ) have no nonzero terms with negative exponents. Then also sums and products of these S_i(T ) have no nonzero terms with negative exponents, implying that the Pk(T )/P0(T ) have no nonzero terms with negative exponents. This contradicts the irreducibil- ity of F , so we conclude that such a S_i(T ) must exist.

2.3 Local Rings and Valuations

In Chapter Modern Painlev´e Theory the tools for studying differential equations locally will be local rings and valuations. This section briefly discusses these notions together with some additional terminology. In what follows assume that K ⊃ C is a field extension (later we take this to be a differential field extension).

Definition 2.8. A valuation ring over C of a field K ⊃ C is a ring O such that:

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V1: C ( O ( K;

V2: a ∈ K\O =⇒ a⁻¹ ∈O.

One can show that a valuation ring O is a local ring. That is, it has a unique maximal ideal. These maximal ideals play an important role, and therefore have their own name.

Definition 2.9. A maximal ideal of a valuation ring of K over C is called a place over C of K.

Remark 2.10. The terminology place comes from the fact that the places of a function field correspond to the points on the corresponding curves.

Remark 2.11. A place determines its corresponding local ring. If p is a place of K ⊃ C, then its local ring is given by Op := {z ∈ K | z⁻¹ ∈ p}./ Furthermore, places in an algebraic function field correspond one-to-one to the points on the corresponding algebraic curve. When an element f of the function field K is not an element of this local ring Op, we say that p is a pole of f .

Definition 2.12. Let p be a place in the field K ⊃ C. The field Kp :=Op/p is called the residue (class) field of p. The degree of a place p is defined as deg p := [Kp: C]. Places of degree one are called C-rational points.

The following lemma is also known as the valuative criterion of proper- ness. It is a duality between places and discrete valuation rings, and we shall need it in Chapter Modern Painlev´e Theory. For a proof, see for instance [7, Chapter 7, Theorem 1, Corollary 4] or [9, Chapter 1, Corollary 6.6].

Lemma 2.13. Let X be a nonsingular projective curve and K ⊃ C its function field. Then there is a one-to-one correspondence between the points of X and the discrete valuation rings of K ⊃ C. If x ∈ X, then Ox=OX,x

is the corresponding discrete valuation ring.

We shall use the following lemma over and over again. It tells us that we can identify the local ring of a nonsingular point with some subring of the convergent power series. This will enable us to make explicit calculations in local rings.

Lemma 2.14. Let Ox be a local ring corresponding to a nonsingular point x of a curve X, and let π be a local parameter at x. The mapping

Ox−→ C{π}, f 7−→

∞

X

n=0

anπⁿ,

where the an are defined by f −Pk

n=0anπⁿ∈ (π^k+1) for all k ∈ Z≥0, is an inclusion.

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Proof. See for instance [20, Chapter II.2.2, Theorem 5].

If we take the field of fractions of a local ring Op, then we obtain the function field. In other words, if z /∈ Op then z⁻¹ ∈ Op. Once we have such a power series representation of the elements of Op, we can therefore extend it to a Laurent series representation of the whole function field. More precisely, we have the following corollary.

Corollary 2.15. Let z₀ be a point in C that is unramified (see below) in some finite field extension C ⊃ C(z). Then there exists an embedding C ,→

C({z − z0}).

2.4 Ramification and Branch Points

In this section we shall talk about the local structure of dense morphisms between curves, and in particular redefine branch points. The following theorem makes this local structure explicit.

Theorem 2.16. For every dense morphism f : X → Y of nonsingular curves over C, and for every x ∈ X, there exist neighborhoods U 3 x and V 3 f (x) and homeomorphisms u : U → C and v : V → C onto neighborhoods of 0 in C such that the diagram

U C

V C

...u . ......................

.... ......................

v

...

. .. .. .......

f

...

. .. .. .......

ρ_k

is commutative. Here ρ_k(z) = z^k, where k is defined as the order of the zero of the function f^∗(t) at x, for t a local parameter at f (x).

Proof. See [21, Chapter VII.3.1, p. 131-3].

This theorem shows us that every dense morphism locally looks like the map z 7→ z^k, for a certain k ∈ N. This leads to the following definition.

Definition 2.17. The number k from the previous theorem is called the ramification degree of f at x ∈ X. If for some x ∈ f⁻¹(y) the ramification degree of f at x is greater than 1, then y is called a branch point or ramification point of f .

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2.5 Differential Function Fields

In the classical theory of differential equations, the basic object was always an equation of the form F (z, y⁰, y) = 0. Furthermore, solutions y were always real or complex-valued functions defined on some domain of R or C. In this section we shall define a new object, called a differential function field, that will in Chapter Modern Painlev´e Theory replace the classical notion of a differential equation.

In the following let A be a ring and let B be an A-algebra, both commutative, with unity and without zerodivisors.

Definition 2.18. A mapping d : B → B is called an A-derivation if it satisfies the following conditions for all x ∈ A and all y, z ∈ B:

D1: d(y + z) = d(y) + d(z);

D2: d(yz) = yd(z) + d(y)z;

D3: d(x) = 0.

It is also called an A-derivation on B. In many cases we shall denote the derivation of an element x simply by x⁰.

Proposition 2.19. Under the assumption that B has no zerodivisors, we can extend a derivation d : B → B to a derivation on the quotient field K of B in one and only one way.

Proof. Uniqueness: Let bd : K → K be an arbitrary derivation on K equal to the derivation d on B. Then for each x ∈ B we have that

1 = bd(x · 1

x) = x bd(1 x) + 1

xd(x)

implying that for all x, y ∈ B, y 6= 0, we have d(bx

y) = 1

yd(x) − x

y²d(y). (2.1)

The derivation bd is therefore determined once it is defined on B.

Existence: If we define bd : K → K by formula 2.1, then it is a straightforward calculation to show that the axiom’s D1 and D2 of a derivation are satisfied.

Several notions from field theory generalize directly to notions with an additional derivation structure. More precisely, we have the following definitions.

Definition 2.20. A differential ring is a pair (R, d), where R is a ring and d : R → R is a derivation. Sometimes we shall denote the differential field by R for short. If R is actually a field, then we call R a differential field .

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Definition 2.21. An element of a differential field whose derivative vanishes is called a constant . The set of all constants in a differential field is called the field of constants.

It is straightforward to show that the field of constants is indeed a field.

It is a trivial example of a differential subfield.

Definition 2.22. A subfield K ⊂ L of a differential field is called a differential subfield of L if it is closed under the differentiation on L. The field extension L ⊃ K together with an extension of the deriva is called a differential field extension of K.

Definition 2.23. An extension field K ⊃ C is called an algebraic function field of one variable over C if there is an element x ∈ K that is transcendental over C, and K is algebraic of finite degree over C(x). It will be called a differential (algebraic) function field of one variable over C if C is a differential subfield of K.

With these new notions of differential rings and differential fields come new questions about how morphisms between them behave, in particular under extensions of the fields.

Lemma 2.24. Suppose φ : R₁ ,→ R₂ is an inclusion of differential rings, where R₁ and R₂ are commutative and have no zerodivisors. Then φ can be extended uniquely to an inclusion bφ : F1 ,→ F2 of differential fields, where F_i is the field of fractions of R_i.

Proof. Uniqueness: Suppose we have a homomorphism bφ : F₁ → F₂ of differential fields that restricts to φ on R1. Then, for any x ∈ R1, we have that 1 = φ(x ·¹_x) = φ(x) · bφ(¹_x) implies that bφ is uniquely determined by the formula

φ(x

y) = φ(x) · φ(y)⁻¹, for all x

y ∈ F₁ (2.2)

and thereby by its definition on R₁.

Existence: Define φ : F1 → F₂ by equation 2.2. This can be done since φ(y) = 0 implies that y = 0. It is straightforward to show that this is an inclusion of differential fields that restricts to φ on R₁.

Lemma 2.25. Let d : K → L be a derivation, and suppose that 0 6= x ∈ L is algebraic over K. Then d can be extended to a derivation bd : K(x) → L in one and only one way.

Proof. Uniqueness: Assume we have such a derivation bd, and let f (X) = Pn

k=0a_kX^k∈ K[X] be the minimal polynomial of x. Then 0 = d(f (x)) =

n

X

k=0

d(ak)x^k+

n

X

k=1

kakx^k−1d(x),b

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and consequently

d(x) = −b Pn

k=0d(ak)x^k Pn

k=1ka_kx^k−1, (2.3)

which is well-defined because every minimal polynomial is separable when char(K) = 0. It follows that the derivation bd is determined by d.

Existence: Define bd : K(x) → L by formula 2.3. Then it is a straightforward calculation to show that bd equals d on K and that it satisfies D1 and D2.

Theorem 2.26 states an analogous result for a transcendental extension.

The following lemma shows how an algebraic function field of one variable over a given differential field C becomes a differential extension of C.

Theorem 2.26. Let K be an algebraic function field of one variable over a given differential field C. Let x be a separating variable in K. For any element y ∈ K there is a unique derivation bd : K → K such that bd(x) = y and it coincides with the given derivation d : C → C.

Proof. If φ is an element of C[x] of the formP a_ixⁱ, then we define d(φ) =b X

d(ai)xⁱ+ yX

iaixⁱ⁻¹.

Clearly it is the unique derivation of C[x] into K that satisfies d(x) = y,b d(a) = d(a),b for all a ∈ C.

By the arguments of Proposition 2.19, bd can be extended to C(x) in a unique way. Since K is separably algebraic over C(x), Lemma 2.25 implies that bd can be extended in a unique way to a derivation on K.

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CHAPTER 3 Classical Painlev´e Theory

I

n this thesis we examine Painlev´e Theory from two points of view: the classical function-theoretic approach and a modern differential algebro- geometric approach. First of all we shall in this chapter consider Painlev´e Theory from the classical point of view.

In Section 3.1, we shall give an overview of several primary and secondary sources in the classical literature concerning first order equations with the PP. After this, we shall discuss two of the most important theorems appear- ing in the primary sources, starting in Section 3.2 with a theorem of Briot and Bouquet for autonomous equations. We shall restate this theorem and its proof in a more readable way, and show that it generalizes the result we found in Chapter Introduction for equations of the form y^0p = y^q. In Section 3.3 we shall turn this theorem into an algorithm that decides if an autonomous first order equation has the PP. Finally, in Section 3.4, we shall discuss the generalization of the theorem of Briot and Bouquet to general first order equations, called Fuchs’s Criterion.

3.1 First Order Painlev´ e Theory in the Literature

This section is concerned with the appearance of first order Painlev´e Theory in the literature. It is often hard to say when a historical development begins, and the same holds for Painlev´e Theory. One could argue that it started at the moment that multivalued functions were understood better, functions that strictly speaking do not live on the complex plane but on some covering space. This was at the time of Puiseux.

Victor Alexandre Puiseux was the first to make the distinction between poles, essential singularities and branch points [2, p. 571]. Furthermore, he had made a theorem that can be stated nowadays as that the algebraic

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(a) The French mathematician Victor Alexandre Puiseux (1820-1883).

(b) The French mathematician Jean Claude Bouquet (1819-1885).

Figure 3.1: Two 19th century mathematicians who layed the foundations for Painlev´e Theory.

closure of the field of (convergent) Laurent series is the field of (convergent) Puiseux series [19]. His work on these functions cleared the road for a better understanding of multivalued functions and thereby a better understanding of multivalued solutions of differential equations.

Briot and Bouquet, who knew Puiseux from the École Normale Supérieure, studied autonomous differential equations of the form F (y⁰, y) = 0, where F is a rational function in y⁰ and y. They asked themselves when such equations have a solution with a branched point. In 1856 they published three important articles in the Journal de l’ École (Imperiale) Polytechnique.

The first one contained some notions from the theory of complex variables, the second was about differential equations in the complex domain and the third about the integration of such differential equations by means of elliptic functions. In this last article, l’Intégration des équations différentielles au moyen des fonctions elliptiques [3], they answered the question when such a differential equation has a solution with a branch point. More precisely, they state and prove one way (the necessity of the conditions) of the following theorem and remark that the other way is trivial.

Theorem 3.1 (Briot and Bouquet). Pour qu’une ´equation diff´erentielle du premier ordre de la forme

du dz

m

+ f1(u) du dz

m−1

· · · + f_m(u) = 0

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admette une intégrale monodrome: 1^◦ les coefficients f₁(u), f₂(u), . . . , f_m(u) doivent être des polynômes entiers en u et, au plus, le premier du second degré, le second du quatrième degré,. . . ,le dernier du degré 2m; 2^◦ quand, pour une certaine valeur de u, l’équation a une racine multiple différente de zéro, ^du_dz doit rester monodrome par rapport à u; 3^◦ quand, pour une certaine valeur u1 de u, l’équation a une racine multiple égale à zéro, le premier terme du développement de ^du_dz, suivant les puissances croissantes de (u − u₁)¹ⁿ, doit avoir l’exposant ⁿ⁻¹_n , si cet exposant est plus petit que l’unité; 4^◦ enfin l’équation différentielle que l’on déduit de la première en posant u = ¹_v, doit offrir, pour v = 0, les mêmes caractères.

They then used this theorem to carry out a detailed analysis of the binomial equations ^du_dzm

= f (u), with f a polynomial in u.

In Section 3.2 we shall rephrase this theorem in a more readable form and prove it in a classical way. As we can see, the criterion is technical and does not (directly) give us much insight. It is, however, correct, and in Section 3.3 we turn this theorem into an algorithm to determine if such an autonomous first order equation has the PP.

Something else that was already found by Briot and Bouquet is the classification of all autonomous equations with the PP by the genus of the associated curve, into the Riccati and the Weierstrass equations. Altogether we can say that Briot and Bouquet completely solved the autonomous case, thereby laying the foundation for the general case.

The next person who made a contribution was Lazarus Immanuel Fuchs (who should not be confused with his less famous son R. Fuchs, who con- tributed to the second order equations with the PP). In 1884, Fuchs tried to generalize Briot and Bouquets theorem to nonautonomous differential equations but made some mistakes [6]. More precisely, he makes the following claim.

Claim 3.2 (Fuchs). Die nothwendigen und hinreichenden Bedingungen daf¨ur, dass die Integrale der Gleichung F (z, y⁰, y) = 0 feste, sich nicht mit den ¨Anderungen der Anfangswerthe stetig verschiebende Verzweigungspunkte besitzen, sind die folgenden:

1. Die Gleichung hat die Form:

y^0m+ ψ1y^0m−1+ ψ2y^0m−2+ · · · + ψm = 0,

worin ψ₁, ψ₂, . . . , ψ_m ganze rationale Functionen von y mit von z ab- h¨angigen Coefficienten von der Beschaffenheit bedeuten, dass ψkh¨och- stens vom Grade 2k in Bezug auf y ist.

2. Ist y = η eine Wurzel der Discriminantengleichung (C.), f¨ur welche die durch (F.) definirte algebraische Function y⁰ von y sich verzweigt,

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(a) The French mathematician Henri Poincar´e (1854- 1912).

(b) The German mathematician Lazarus Immanuel Fuchs (1833-1902).

Figure 3.2: Two mathematicians who solved the general case for first order equations.

so ist η ein Integral der Gleichung (F.). In der y⁰ als algebraische Func- tion von y darstellenden Riemannschen Fl¨ache hat y⁰ in s¨amtlichen

¨

uber y = η liegenden Verzweigungsstellen den Werth y⁰ = ζ = ^dη_dz. 3. Je α Bl¨attern, welche sich in y = η, y⁰ = ζ = ^dη_dz verzweigen, entsprechen

mindestens α − 1 mit y = η zusammenfallende Wurzeln der Gleichung F (z, y, ζ)

mit der Unbekannten y.

This criterion determines whether a first order differential equation of the form F (z, y⁰, y) = 0, F rational in y⁰ and y, has the PP. It is strange that he does not specify its dependence on z, because it is in this that he generalizes the theorem of Briot and Bouquet. One could for instance take F to be algebraic in z. Although all the conditions in the criterion are necessary, they are altogether not sufficient. It is strange that Fuchs made this mistake, because it is easy to think of examples of functions that do pass Fuchs’s test but do not pass the test of Briot and Bouquet. As we shall see in Section 3.4 the criterion can be made sufficient by adding an extra condition, thus obtaining what is nowadays known as Fuchs’s Criterion.

Poincar´e’s response to Fuchs’s article was an article published in 1885 in which he considered the case of a genus bigger than two [18]. After this, Paul Painlev´e published an article in 1885 [16] on the first order case, combined all previously obtained results together with his new results for the second

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order equations into his Stockholm Lectures [17] (that appeared in 1895), thereby creating Painlevé Theory. But Painlevé Theory is a tricky business, because once again many mistakes (at least for the second order case) were made in this work. The classification of the second order equations into the six Painlevé equations known nowadays was completed by his student B.

Gambier, but the proof that all these equations do have the PP themselves was done only a few years ago by Masahiko Saito et al. at Kobe University.

So the Painlevé Theory constructed so far was far from perfect. In 1927 the British mathematician Edward Lindsay Ince wrote a book in which he summed up the most important results and techniques developed so far for ordinary differential equations in a book called Ordinary Differential Equa- tions [11]. Although the book became a classic and a standard reference, its section about first order Painlevé Theory is incomplete, imprecise and sometimes wrong. Despite these shortcomings, many mathematicians take this book to be the standard reference concerning first order Painlevé Theory.

In 1909, the Swedish mathematician Axel Johannes Malmquist (1882 - 1952) wrote his dissertation called Sur les équations différentielles du premier ordre dont l’integrale generale admet un nombre fini de branches per- mutable autor des points critiques mobiles under the supervision of Gösta Mittag-Leffler. This dissertation was followed by an article in 1913 on the same topic [13], and then no further work was published for many years. In 1940, he all of a sudden started publishing results again. He concluded this publishing spree with an article in which he gives the classification of the first order equations with the PP in a very precise form, but the proofs are very classical and therefore very difficult to understand [14].

In a next attempt to sum up what exactly had been done in the area of first order equations with the PP, the German mathematician and Nazi Ludwig Georg Elias Moses (!) Bieberbach wrote a book in 1953, titled Theorie der gew¨ohnlichen Differentialgleichungen: auf funktionentheoretis- che Grundlage dargestellt, in which he recited several classical statements and proved some of them [1]. These statements and proofs are much more precise than what was done in Ince’s book.

In 1976, the Swedish mathematician Einar Carl Hille wrote the book Or- dinary Differential Equations in the Complex Domain [10] simply because, as he stated in the preamble, there was at that moment not yet a book dedi- cated solely to this topic. In quite a large part of the book he considers first order equations with the PP and proves in a very precise and readable way results about the explicit differential equations y⁰ = R(z, y), R a rational function of z and y.

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(a) The German mathematician and Nazi Ludwig Georg Elias Moses Bieberbach (1886- 1982).

(b) The Swedish mathematician Einar Carl Hille (1895- 1980).

Figure 3.3: Two mathematicians who redid some classical Painlev´e Theory.

3.2 Classical Proof of the Theorem of Briot and Bouquet

In this section we shall restate and reprove the theorem of Briot and Bouquet that we encountered in Section 3.1. We start with some remarks that might help in the understanding of the proof of the theorem.

Remark 3.3. In what follows, the reader should be careful and realize that we work with two types of Puiseux series: those of the formP

kaky^k/d that help to factorize the differential equation and those of the formP

ka_kz^k/e that denote solutions of the differential equation.

Remark 3.4. We first remark that the equation not having a solution with a movable branch point is equivalent to the equation not having a solution with a branch point at 0. To see this, suppose that the equation has a solution w with a branch point at 0. Then, since the differential equation is autonomous, the function

Φ : (z, t) 7−→ w(z + c(t))

has the PP for a sufficiently small neighborhood U of zero and every noncon- stant path c : [0, 1] → U with c(0) = 0. On the other hand, if a differential equation has a solution with a movable branch point then it has of course a solution with a branch point at 0 as well.

Within the proof of the theorem of Briot and Bouquet, we make use of the following lemmas several times.

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Lemma 3.5. Suppose we have a function w : U → C, where U is a neighborhood of z0, that can be expressed as a convergent power series of the form

w − w0=

∞

X

k=1

w_k(z − z0)^k, w1 6= 0.

Then the function w − w0 has an inverse z − z0, defined in a neighborhood of 0, that can be expressed as a convergent power series as well, given by

z − z₀= 1

w₁(w − w₀) +

∞

X

k=2

z_k(w − w₀)^k.

Proof. It follows from the complex version of the implicit function theorem that there exists locally an inverse that can be expressed as a convergent power series. Composing power series shows that it must be of the given form.

Lemma 3.6. Suppose we have a function w : U → C, where U is a neighborhood of z₀, that can be expressed as a convergent Puiseux series of the form

w − w0=

∞

X

k=n

wk(z − z0)^k/d, wn6= 0, n ≥ 1, d ≥ 1.

Then the function w − w0 has an inverse z − z0, defined in a neighborhood of 0, that can be expressed as a convergent Puiseux series of the form

z − z₀=

"_∞ X

k=1

z_k(w − w₀)^k/n

#d

, z₁ 6= 0.

Proof. Because we can write every power series with nonzero constant term as an arbitrary power of some other power series with nonzero constant term, we find that

w − w₀= (z − z₀)^n/d

"_∞ X

k=0

w⁰_k(z − z₀)^k/d

#n

, w₀⁰ 6= 0, n ≥ 1, for a certain convergent Puiseux series between the brackets. Taking on each side the n⁻¹th powers, we find that (w − w0)^1/n as a function of (z − z0)^1/d has (by Lemma 3.5) an inverse whose dth power is of the form proposed in the lemma.

Lemma 3.7. Suppose w : U → V and its inverse (under composition) z : V → U are functions on certain domains U, V ⊂ P that have expansions as a Puiseux series in their whole domain. If w is a solution to ^dX_dt = F (X) such that F (w(z)) 6= 0 for every z ∈ U , then its inverse (under composition) z is a solution to ^dX_dt = _{F (t)}¹ .

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Proof. Using the chain rule in the differential field of convergent Puiseux series, we find that

1 ≡ d(IdV)

dt (t) = d(w ◦ z)

dt (t) = dw

dz(z(t)) ·dz

dt = F (t) ·dz dt, from which the lemma follows since F (t) 6= 0 for every t ∈ V .

Now we are prepared to state and prove the theorem of Briot and Bou- quet.

Theorem 3.8 (Briot and Bouquet, 1856). Consider the differential equation

F (y⁰, y) = P₀(y)y^0m+ · · · + P_m(y) = 0, F ∈ C[S, T ] (3.1) irreducible with S-discriminant ∆(T ), P₀ 6≡ 0. For every a ∈ C, we can write F (S, T ) in a unique way in the form

F (S, T ) = P0(T )

m

Y

i=1

(S − Si(T )) , Si∈ C({(T − a)^1/d}).

The differential equation has no solution with a movable branch point if and only if the following criteria for F hold:

A1: For k = 0, . . . , m we have that deg Pk ≤ 2k.

A2: If, for a certain a ∈ C, ∆(a) = 0 and one of the Si(T ) has a nonzero constant term, then this S_i(T ) has only terms with integral exponents.

A3: If, for a certain a ∈ C, ∆(a) = 0 and one of the Si(T ) has a zero constant term and starts at a term with an exponent less than one, then this S_i(T ) starts at a term of the form (T − a)^(d−1)/d.

A4: If we transform the equation F (y⁰, y) = 0 by y = 1/by and y⁰ = −yb⁰/yb², then we obtain an equation of the form

F (b by⁰,by) = bP₀(y)b by^0m+ · · · + bP_m(by) = 0, F ∈ C[S, T ]b (3.2) irreducible with S-discriminant b∆(T ), bP0 6≡ 0. For every a ∈ C, we can write bF (S, T ) in a unique way in the form

F (S, T ) = bb P_0,a(T − a)

m

Y

i=1

S − bS_i(T )

, Sb_i ∈ C({(T − a)^1/d}).

The criteria A2 and A3 hold for a = 0, if we omit the wide hats.

The following proof is a mod(ern)ification of the original proof by Briot and Bouquet, given in their article [3].

The Classification of the First Order Ordinary Differential Equations with the Painlevé Property

The Classification of the First Order Ordinary Differential Equations with the Painlev´e Property

Preface and Acknowledgments

CONTENTS

LIST OF FIGURES

CHAPTER 1 Introduction

T

1.1 Problematic Points

1.2 The Painlev´ e Property

1.3 First Order Equations with the PP

CHAPTER 2 Prerequisites

I

2.1 Fields of Functions

2.2 Factorizing a Polynomial Differential Equation

2.3 Local Rings and Valuations

2.4 Ramification and Branch Points

2.5 Differential Function Fields

CHAPTER 3

Classical Painlev´e Theory

I

3.1 First Order Painlev´ e Theory in the Literature

3.2 Classical Proof of the Theorem of Briot and Bouquet