**Mathematical Surveys and Monographs**

Volume: 249;
2020;
187 pp;
Softcover

MSC: Primary 11;
Secondary 05; 12; 20

**Print ISBN: 978-1-4704-5634-4
Product Code: SURV/249**

List Price: $140.00

AMS Member Price: $112.00

MAA Member Price: $126.00

**Electronic ISBN: 978-1-4704-6029-7
Product Code: SURV/249.E**

List Price: $140.00

AMS Member Price: $112.00

MAA Member Price: $126.00

#### Supplemental Materials

# Davenport–Zannier Polynomials and Dessins d’Enfants

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*Nikolai M. Adrianov; Fedor Pakovich; Alexander K. Zvonkin*

The French expression “dessins d'enfants”
means children's drawings. This term was coined by the great French
mathematician Alexandre Grothendieck in order to denominate a method
of pictorial representation of some highly interesting classes of
polynomials and rational functions. The polynomials studied in this
book take their origin in number theory. The authors show how, by
drawing simple pictures, one can prove some long-standing conjectures
and formulate new ones. The theory presented here touches upon many
different fields of mathematics.

The major part of the book is quite elementary and is easily
accessible to an undergraduate student. The less elementary parts,
such as Galois theory or group representations and their characters,
would need a more profound knowledge of mathematics. The reader may
either take the basic facts of these theories for granted or use our
book as a motivation and a first approach to these
subjects.

#### Readership

Graduate students and researchers interested in learning about combinatorics of polynomials as part of the new theory of dessins d'enfants.

#### Table of Contents

# Table of Contents

## Davenport-Zannier Polynomials and Dessins d'Enfants

- Cover Cover11
- Title page iii5
- Copyright iv6
- Contents v7
- Preface vii9
- Chapter 1. Introduction 115
- Chapter 2. Dessins d’enfants: from polynomials through Belyĭ functions to weighted trees 923
- Chapter 3. Existence theorem 1933
- Chapter 4. Recapitulation and perspectives 2741
- Chapter 5. Classification of unitrees 2943
- 5.1. Statement of the main result 2943
- 5.2. Weight distribution 3650
- 5.3. Brushes 3953
- 5.4. Trees with repeating branches of height 2 4155
- 5.5. Trees with repeating branches of the type (1,𝑠,𝑠+1) 4559
- 5.6. Trees with repeating branches of the type (1,𝑡,1) 5064
- 5.7. The trees 𝐴, 𝐵, …, 𝑇 listed in Theorem 5.4 are unitrees 5367
- 5.8. Appendix: the inverse enumeration problem 5670

- Chapter 6. Computation of Davenport–Zannier pairs for unitrees 5973
- 6.1. Reciprocal polynomials 5973
- 6.2. Remarks about computation 6074
- 6.3. Stars and binomial series 6175
- 6.4. Forks and Hall’s conjecture 6276
- 6.5. Jacobi polynomials 6478
- 6.6. Series 𝐹 and 𝐺: trees of diameter 4 7286
- 6.7. Series 𝐻 and 𝐼: decomposable ordinary trees 7791
- 6.8. Series 𝐽 8094
- 6.9. Sporadic trees 8296

- Chapter 7. Primitive monodromy groups of weighted trees 87101
- Chapter 8. Trees with primitive monodromy groups 99113
- Chapter 9. A zoo of examples and constructions 137151
- 9.1. Composition 137151
- 9.2. Difference of powers over Q : infinite series 140154
- 9.3. Polynomials with a relaxed minimum degree condition 141155
- 9.4. Duality and self-duality 143157
- 9.5. A “historic” sporadic example 146160
- 9.6. Some sporadic examples of Beukers and Stewart 148162
- 9.7. Sporadic examples of “megamap invariant” 152166
- 9.8. One more application due to David Roberts 154168

- Chapter 10. Diophantine invariants 155169
- Chapter 11. Enumeration 163177
- Chapter 12. What remains to be done 175189
- Bibliography 179193
- Index 185199
- Back Cover Back Cover1202