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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.

Davenport-Zannier Polynomials and Dessins d'Enfants

Introduction.

Dessins d'enfants: From polynomials through Belyi functions to weighted trees.

Existence theorem.

Recapitulation and perspective.

Classification of unitrees.

Computation of Davenport-Zannier pairs for unitrees.

Primitive monodromy groups of weighted trees.

Trees with primitive monodromy groups.

A zoo of examples and constructions.

Diophantine invariants.

Enumeration.

What remains to be done.

Bibliography.

Index.