Dynamical Systems VIII: Singularity Theory II. Applications

240 Pages·1993·6.482 MB·English

by  V. I. Arnol’d, V. V. Goryunov, O. V. Lyashko, V. A. Vasil’ev (auth.), V. I. Arnol’d (eds.)

Encyclopaedia of
Mathematical Seiences
Valurne 39
Editor-in-Chief: R.V. Gamkrelidze
V. I. Arnol'd (Ed.)
Dynamical Systems VIII
Singularity Theory II
Applications
With 134 Figures
Springer-Verlag Berlin Heidelberg GmbH
Consulting Editors of the Series:
A.A. Agrachev, A.A. Gonchar, E.F. Mishchenko,
N.M. Ostianu, V.P. Sakharova, A.B. Zhishchenko
Title of the Russian edition:
ltogi nauki i tekhniki, Sovremennye problemy matematiki,
Fundamental'nye napravleniya, Vol. 39,
Dinamicheskie Sistemy 8
Publisher VINITI, Moscow 1989
Mathematics Subject Classification (1991):
32Sxx, 34Cxx, 35Bxx, 35Qxx, 49Lxx, 51-XX, 53Cxx,
57R45, 58C27, 58Fxx, 58Gxx, 70Hxx, 93-XX
Library of Congress Cataloging-in-Publication Data
Dynamical systems. (Encyclopaedia of mathematical sciences; v. I
Translation of: Dinamicheskie sistemy, issued as part of the series: Itogi nauki i tekhniki.
Seriiä Sovremennye problemy matematiki. Vol. 3 edited by V. I. Arnold.
Includes bibliographies and indexes.
Contents: I. Ordinary differential equations and smooth dynamical systems/
D.V. Anosov, V. I. Arnold (eds.)-2. Ergodie theory with applications to dynamical systems
and statistical mechanics(Ya.G. Sinai (ed.)- -8. Singularity theory 2.
I. Mechanics, Analytic. 2. Mechanics, Celestial.
I. Anosov, D.V. II. Arnol'd, V.l. (Vladimir Igorevich), 1937-
III. Series: Encyclopaedia ofmathematical sciences; v. I, etc.
QA805.D5613 1988 531 87-20655
ISBN 978-3-642-08101-9 ISBN 978-3-662-06798-7 (eBook)
DOI 10.1007/978-3-662-06798-7
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©Springer-Verlag Berlin Beideiberg 1993
Originally published by Springer-V erlag Berlin Heidelberg N ew Y ork in 1993
Softcover reprint of tbe bardeover 1st edition 1993
Typesetting: Asco Trade Typesetting Ltd., Hong Koog
41/3140-543210-Printed on acid-free paper
List of Editors, Authors and Translators
Editor-in-Chief
R.V. Gamkrelidze, Russian Academy of Sciences, Steklov Mathematical Insti
tute, ul. Vavilova 42, 117966 Moscow, Institute for Scientific Information
(VINITI), ul. Usievicha 20a, 125219 Moscow, Russia
Consulting Editor
V.I. Arnol'd, Steklov Mathematical Institute, ul. Vavilova 42, 117966 Moscow,
Russia
Authors
V.I. Arnol'd, Steklov Mathematical Institute, ul. Vavilova 42, 117966 Moscow,
Russia
V.V. Goryunov, Moscow Aviation Institute, Volokolamskoe sh. 4,
125871 Moscow, Russia
O.V. Lyashko, Airforce Engineering Academy, Leningradskij pr. 40,
125167 Moscow, Russia
V.A. Vasil'ev, Institute of Applied Mathematics of the Russian Academy of
Sciences, Miusskaya pl. 4, 125047 Moscow, Russia
Transtator
J.S. Joel, P.O. Box 132, Kelly, WY 83011, USA
Singularity Theory II
Classification and Applications
V.I. Arnol'd, V.V. Goryunov, O.V. Lyashko, V.A. Vasil'ev
Translated from the Russian
by J.S. Joel
Contents
Foreword 6
Chapter 1. Classification of Functions and Mappings . . . . . . . . . . . . . . . . 8
§ 1. Functions on a Manifold with Boundary . . . . . . . . . . . . . . . . . . . . . . . 8
1.1. Classification of Functions on a Manifold with a Smooth
Boundary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.2. Versal Deformations and Bifurcation Diagrams . . . . . . . . . . . . . 11
1.3. Relative Homology Basis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
1.4. Intersection Form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
1.5. Duality of Boundary Singularities . . . . . . . . . . . . . . . . . . . . . . . . . 17
1.6. Functions on a Manifold with a Singular Boundary . . . . . . . . . . 17
§ 2. Complete Intersections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.1. Start ofthe Classification .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. . 21
2.2. Critical and Discriminant Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
2.3. The Nonsingular Fibe~ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
2.4. Relations Between the Tyurina and Milnor Numbers . . . . . . . . 28
2.5. Adding aPower of a New Variable . . . . . . . . . . . . . . . . . . . . . . . 29
2.6. Relative Monodromy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
2.7. Dynkin Diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
2.8. Parabolic and Hyperbolic Singularities . . . . . . . . . . . . . . . . . . . . 31
2.9. Vector Fields on a Quasihomogeneaus Complete Intersection 33
2.10. The Space of a Miniversal Deformation of
a Quasihomogeneaus Singularity . . . . . . . . . . . . . . . . . . . . . . . . 35
2.11. Topological Triviality of Versal Deformations . . . . . . . . . . . . . . 36
§ 3. Projections and Left-Right Equivalence . . . . . . . . . . . . . . . . . . . . . . . . 37
3.1. Projections of Space Curves onto the Plane . . . . . . . . . . . . . . . . . 38
3.2. Singularities of Projections of Surfaces onto the Plane . . . . . . . . 39
3.3. Projections of Complete Intersections . . . . . . . . . . . . . . . . . . . . . . 43
2 Contents
3.4. Projections onto the Line . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
3.5. Mappings of the Line into the Plane . . . . . . . . . . . . . . . . . . . . . . . 57
3.6. Mappings of the Plane into Three-Space . . . . . . . . . . . . . . . . . . . 59
§ 4. N onisolated Singularities of Functions . . . . . . . . . . . . . . . . . . . . . . . . . 65
4.1. Transversal Type of a Singularity . . . . . . . . . . . . . . . . . . . . . . . . . 65
4.2. Realization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
4.3. Topology of the Nonsingular Fiber . . . . . . . . . . . . . . . . . . . . . . . . 66
4.4. Series of Isolated Singularities . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
4.5. The Number oflndices of a Series . . . . . . . . . . . . . . . . . . . . . . . . . 68
4.6. Functions with a One-Dimensional Complete Intersection as
Critical Set and with Transversal Type A 1 . . • • . . . • . . . . . . . . . . 69
§ 5. Vector Fields Tangent to Bifurcation Varieties . . . . . . . . . . . . . . . . . . 79
5.1. Functions on Smooth Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . 79
5.2. Projections onto the Line . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
5.3. Isolated Singularities of Complete Intersections . . . . . . . . . . . . . 82
5.4. The Equation of a Free Divisor . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
§ 6. Divergent and Cyclic Diagrams of Mappings . . . . . . . . . . . . . . . . . . . 84
6.1. Germs of Smooth Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
6.2. Envelopes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
6.3. Holomorphic Diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
Chapter 2. Applications of the Classification of Critical Points of
Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
§ 1. Legendre Singularities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
1.1. Equidistants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
1.2. Projective Duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
1.3. Legendre Transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
1.4. Singularities of Pedalsand Primitives . . . . . . . . . . . . . . . . . . . . . . 91
1.5. The Higher-Dimensional Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
§ 2. Lagrangian Singularities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
2.1. Caustics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
2.2. The Manifold of Centers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
2.3. Caustics of Systems of Rays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
2.4. The Gauss Map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
2.5. Caustics of Potential Systems of Noninteracting Particles . . . . . 95
2.6. Coexistence of Singularities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
§ 3. Singularities of Maxwell Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
3.1. Maxwell Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
3.2. Metamorphoses of Maxwell Sets . . . . . . . . . . . . . . . . . . . . . . . . . . 100
3.3. Extended Maxwell Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
3.4. Complete Maxwell Set Close to the Singularity A5 • • • • . • • • • • • 106
3.5. The Structure of Maxwell Sets Close to the Metamorphosis A 110
5
Contents 3
3.6. Enumeration of the Connected Components of Spaces of
Nondegenerate Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
§ 4. Bifurcations of Singular Points of Gradient Dynamical Systems . . . . 113
4.1. Thom's Conjecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
4.2. Singularities of Corank One . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
4.3. Guckenheimer's Counterexample . . . . . . . . . . . . . . . . . . . . . . . . . 116
4.4. Three-Parameter Families of Gradients . . . . . . . . . . . . . . . . . . . . 117
4.5. Normal Forms ofGradient Systems D4 • . • . • • • . • • • • • • • • . • • • 118
4.6. Bifurcation Diagrams and Phase Portaits of Standard Families . 118
4.7. Multiparameter Families . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
Chapter 3. Singularities of the Boundaries of Domains of
Function Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
§ 1. Boundary of Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
1.1. Domains of Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
1.2. Singularities of the Boundary of Stability in
Low-Dimensional Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
1.3. Stabilization Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
1.4. Finiteness Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
§ 2. Boundary of Ellipticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
2.1. Domains of Ellipticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
2.2. Stabilization Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
2.3. Boundaries of Ellipticity and Minimum Functions . . . . . . . . . . . 125
2.4. Singularities of the Boundary of Ellipticity in
Low-Dimensional Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
§ 3. Boundary of Hyperbolicity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
3.1. Domain of Hyperbolicity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
3.2. Stabilization Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
3.3. Local Hyperbolicity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
3.4. Local Properties of Domains of Hyperbolicity . . . . . . . . . . . . . . 129
§ 4. Boundary of the Domain of Fundamental Systems . . . . . . . . . . . . . . . 131
4.1. Domain of Fundamental Systems and the Bifurcation Set . . . . . 131
4.2. Singularities of Bifurcation Sets of
Generic Three-Parameter Families . . . . . . . . . . . . . . . . . . . . . . . . 132
4.3. Bifurcation Sets and Schubert Cells . . . . . . . . . . . . . . . . . . . . . . . 136
4.4. Normal Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
4.5. Duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
4.6. Bifurcation Setsand Tangential Singularities . . . . . . . . . . . . . . . 142
4.7. The Group ofTransformations ofSets and Finite Determinacy. 143
4.8. Bifurcation Diagrams of Flattenings of Projective Curves . . . . . 145
§ 5. Linear Differential Equations and Complete Flag Manifolds . . . . . . 146
4 Contents
Chapter 4. Applieations of Ramified Integrals and Generalized Pieard
Lefsehetz Theories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
§ 1. Newton's Theorem on Nonintegrability . . . . . . . . . . . . . . . . . . . . . . . . 150
1.1. Newton's Theorem and Arehimedes's Example . . . . . . . . . . . . . 150
1.2. Multi-dimensional Newton Theorem (Even Case) . . . . . . . . . . . 151
1.3. Obstruetions to Inegrability in the Odd-Dimensional Case . . . 152
1.4. Newton's Theorem for Noneonvex Domains . . . . . . . . . . . . . . . 153
1.5. The Case ofNonsmooth Domains . . . . . . . . . . . . . . . . . . . . . . . 154
1.6. Homologieal Formulation and the General Statement of
the Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154
1.7. Loealization and Lowering the Dimension in the Caleulation
of Monondromy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156
1.8. General Construetion of the Variation Operators . . . . . . . . . . . 157
1.9. The "Cap" Element . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159
1.10. Ramifieation of Cycles Close to Nonsingular Points . . . . . . . . . 160
1.11. Ramifieation Close to Individual Singularities . . . . . . . . . . . . . . 162
1.12. Stabilization of Monodromy Close to Strata of
Positive Dimension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164
1.13. Ramifieation Around the Asymptotie Direetions and
Monodromy of Boundary Singularities . . . . . . . . . . . . . . . . . . . 169
1.14. Pham's Formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171
1.15. Problems, Conjeetures, Complements . . . . . . . . . . . . . . . . . . . . 172
§ 2. Ramifieation of Solutions of Hyperbolie Equations . . . . . . . . . . . . . 173
2.1. Hyperbolie Operatorsand HyperhoHe Polynomials . . . . . . . . 174
2.2. Wave Front of a Hyperbolie Operator . . . . . . . . . . . . . . . . . . . 175
2.3. Singularities of Wave Frontsand Generating Funetions . . . . 175
2.4. Laeunas, Sharpness, Diffusion . . . . . . . . . . . . . . . . . . . . . . . . . . 176
2.5. Sharpness and Diffusion Close to the Simplest Singularities of
Wave Fronts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177
2.6. The Herglotz-Petrovskii-Leray Integral Formula . . . . . . . . . . 178
2. 7. The Petrovskii Criterion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179
2.8. Loeal Petrovskii Criterion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180
2.9. Loeal Petrovskii Cycle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181
2.10. C00-Inversion of the Petrovskii Criterion, Stahle Singularities
of Fronts and Sneaky Diffusion . . . . . . . . . . . . . . . . . . . . . . . . . 183
2.11. Normal Forms of Nonsharpness Close to Singularities of
Wave Fronts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185
2.12. Construetion of Leray and Petrovskii Cycles for Strictly
Hyperbolic Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186
2.13. Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186
§ 3. Integrals of Ramified Forms and Monodromy of Homology with
Nontrivial Coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187
3.1. The Hypergeometrie Funetion of Gauss . . . . . . . . . . . . . . . . . . . 187
Contents 5
3.2. Homology of Local Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189
3.3. Meromorphy of the Integral of the Function pA . . . . . . . . . . . . 191
3.4. The Integral of the Function pA as a Function of P . . . . . . . . . . 194
3.5. Monodromy and Linear Independence of
Hypergeometrie Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197
3.6. Twisted Picard-Lefschetz Theory of Isolated Singularities of
Smooth Functions and Representations of Hecke Algebras 198
Chapter 5. Deformations of Real Singularities and
Local Petrovskii Lacunas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199
§ 1. Local Petrovskii Cycles and their Properties . . . . . . . . . . . . . . . . . . . . 200
1.1. Definition of Local Petrovskii Cycles . . . . . . . . . . . . . . . . . . . . . . 201
1.2. Camplex Conjugation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201
1.3. Boundary of the Petrovskii Class . . . . . . . . . . . . . . . . . . . . . . . . . . 201
1.4. Computation of Petrovskii Cocycles in Terms of
V anishing Cycles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201
1.5. Stabilization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203
§ 2. Local Lacunas for Concrete Singularities . . . . . . . . . . . . . . . . . . . . . . . 204
2.1. Local Lacunas for Singularities that are Stably Equivalent to
Extrema . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204
2.2. The Number of Local Lacunas for the Tabulated Singularities . 204
2.3. Realization of Local Lacunas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208
2.4. Concerning the Proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212
§ 3. Complements of Discriminants of Real Singularities . . . . . . . . . . . . . 212
3.1. Components of the Complement of the Discriminant of
Simple Singularities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212
3.2. A Regular Search Algorithm for Morse Decompositions of
Singularities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213
3.3. Remarks on the Realization of the Algorithm . . . . . . . . . . . . . . . 215
3.4. Problemsand Perspectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216
Bibliography 218
Author Index 231
Subject Index 233
Foreword
In the first volume of this survey (Arnol'd et al. (1988), hereafter cited as "EMS
6") we acquainted the reader with the basic concepts and methods of the theory
of singularities of smooth mappings and functions. This theory has numerous
applications in mathematics and physics; here we begin describing these applica
tions. Nevertheless the present volume is essentially independent of the first one:
all of the concepts of singularity theory that we use are introduced in the course
of the presentation, and references to EMS 6 are confined to the citation of
technical results.
Although our main goal is the presentation of analready formulated theory,
the readerwill also come upon some comparatively recent results, apparently
unknown even to specialists. We pointout some of these results.
In the consideration of mappings from C2 into C3 in§ 3.6 of Chapter 1, we
define the bifurcation diagram of such a mapping, formulate a K(n, 1)-theorem
for the complements to the bifurcation diagrams of simple singularities, give the
definition of the Mond invariant N in the spirit of "hunting for invariants", and
we draw the reader's attention to a method of constructing the image of a
mapping from the corresponding function on a manifold with boundary. In§ 4.6
of the same chapter we introduce the concept of a versal deformation of a
function with a nonisolated singularity in the dass of functions whose critical
sets are arbitrary complete intersections of fixed dimension. The corresponding
K(n, 1)-theorem for simple functions is also true here.
In Chapter 2, following V.l. Bakhtin, we discuss the topology of four vari
ants of the Maxwell variety of the singularity A5, we describe generic "pere
stroikas" of the maximum function under a change of one parameter of the four
and apply this analysis to the study of perestroikas of shock waves propa
gating in a three-dimensional space (following I.A. Bogaevskii); here we give a
"sec+ tan" formula for the nurober of components of the space of Morse func
tions on the line and a formula of A.A. Vakulenko for the nurober of components
of complements to Maxwell strata.
Chapter 3 contains, among other things, classifications of the singularities of
the boundary of the set of hyperbolic differential equations (B.Z. Shapiro and
A.D. Vainshtein) and the singularities of the boundary of the set offundamental
systems of solutions of linear differential equations. (This theory, due to M.E.
Kazaryan, is related to the Schubert stratification of a Grassmannian, to the
bifurcation of Weierstrass points of algebraic curves, and to the theory of the
focal varieties of projective curves.) In the same chapter we discuss the singu
larities of the boundary of the set of disconjugate systems (i.e., Chebyshev
systems)-the connection of this question with the Schubert stratifications of
flag varieties and with Bruhat orderings was recently discovered by B.Z. and
M.Z. Shapiro.
Historically the first result based on the theory of monodromy is Newton's
theorem on the nonintegrability of plane ovals; in §4.1 we shall prove multi-