**Graduate Studies in Mathematics**

Volume: 86;
2008;
625 pp;
Hardcover

MSC: Primary 34;
Secondary 14; 32; 13

**Print ISBN: 978-0-8218-3667-5
Product Code: GSM/86**

List Price: $92.00

AMS Member Price: $73.60

MAA Member Price: $82.80

**Electronic ISBN: 978-1-4704-2116-8
Product Code: GSM/86.E**

List Price: $86.00

AMS Member Price: $68.80

MAA Member Price: $77.40

#### Supplemental Materials

# Lectures on Analytic Differential Equations

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*Yulij Ilyashenko; Sergei Yakovenko*

The book combines the features of a
graduate-level textbook with those of a research monograph and survey
of the recent results on analysis and geometry of differential
equations in the real and complex domain. As a graduate textbook, it
includes self-contained, sometimes considerably simplified
demonstrations of several fundamental results, which previously
appeared only in journal publications (desingularization of planar
analytic vector fields, existence of analytic separatrices, positive
and negative results on the Riemann–Hilbert problem,
Ecalle–Voronin and Martinet–Ramis moduli, solution of the
Poincaré problem on the degree of an algebraic separatrix,
etc.). As a research monograph, it explores in a systematic way the
algebraic decidability of local classification problems, rigidity of
holomorphic foliations, etc. Each section ends with a collection of
problems, partly intended to help the reader to gain understanding and
experience with the material, partly drafting demonstrations of the
more recent results surveyed in the text.

The exposition of the book is mostly geometric, though the algebraic side of
the constructions is also prominently featured. On several occasions the
reader is introduced to adjacent areas, such as intersection theory for
divisors on the projective plane or geometric theory of holomorphic vector
bundles with meromorphic connections. The book provides the reader with
the principal tools of the modern theory of analytic differential equations
and intends to serve as a standard source for references in this area.

#### Readership

Graduate students and research mathematicians interested in analysis and geometry of differential equations in real and complex domain.

#### Reviews & Endorsements

The authors provide a crash course on functions of several complex variables and elements of the theory of Riemann spaces.

-- Mathematical Horizon

The book is easy to read. The ideas and directions are clearly indicated before going into details. ... Moreover, it is easy to open the book at any section and start reading.

-- Mathematical Reviews

#### Table of Contents

# Table of Contents

## Lectures on Analytic Differential Equations

- Cover Cover11 free
- Title iii4 free
- Copyright iv5 free
- Contents vii8 free
- Preface ix10 free
- Chapter I. Normal forms and desingularization 116 free
- 1. Analytic differential equations in the complex domain 116
- 2. Holomorphic foliations and their singularities 1328
- 3. Formal flows and embedding theorem 2944
- 4. Formal normal forms 4055
- 5. Holomorphic normal forms 6176
- 6. Finitely generated groups of conformal germs 8196
- 7. Holomorphic invariant manifolds 105120
- 8. Desingularization in the plane 112127

- Chapter II. Singular points of planar analytic vector fields 143158
- 9. Planar vector fields with characteristic trajectories 143158
- 10. Algebraic decidability of local problems and center-focus alternative 159174
- 11. Holonomy and first integrals 179194
- 12. Zeros of parametric families of analytic functions and small amplitude limit cycles 200215
- 13. Quadratic vector fields and the Bautin theorem 223238
- 14. Complex separatrices of holomorphic foliations 232247

- Chapter III. Local and global theory of linear systems 255270
- 15. General facts about linear systems 255270
- 16. Local theory of regular singular points and applications 265280
- 17. Global theory of linear systems: holomorphic vector bundles and meromorphic connexions 285300
- 18. Riemann–Hilbert problem 312327
- 19. Linear nth order differential equations 329344
- 20. Irregular singularities and the Stokes phenomenon 351366
- Appendix: Demonstration of Sibuya theorem 365380

- Chapter IV. Functional moduli of analytic classification of resonant germs and their applications 373388
- Chapter V. Global properties of complex polynomial foliations 469484
- 25. Algebraic leaves of polynomial foliations on the complex projective plane P[sup(2)] 470485
- Appendix: Foliations with invariant lines and algebraic leaves of foliations from the class A[sup(r)] 499514
- 26. Perturbations of Hamiltonian vector fields and zeros of Abelian integrals 505520
- 27. Topological classification of complex linear foliations 545560
- 28. Global properties of generic polynomial foliations of the complex projective plane P[sup(2)] 567582

- First aid 599614
- Bibliography 607622
- Index 621636
- Back Cover Back Cover1641