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Davenport-Zannier Polynomials and Dessins D'Enfants [Minkštas viršelis]

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  • Formatas: Paperback / softback, 187 pages, aukštis x plotis: 254x178 mm, weight: 369 g
  • Serija: Mathematical Surveys and Monographs
  • Išleidimo metai: 30-Sep-2020
  • Leidėjas: American Mathematical Society
  • ISBN-10: 1470456346
  • ISBN-13: 9781470456344
Kitos knygos pagal šią temą:
Davenport-Zannier Polynomials and Dessins D'EnfantsDidesnis paveikslėlis
  • Formatas: Paperback / softback, 187 pages, aukštis x plotis: 254x178 mm, weight: 369 g
  • Serija: Mathematical Surveys and Monographs
  • Išleidimo metai: 30-Sep-2020
  • Leidėjas: American Mathematical Society
  • ISBN-10: 1470456346
  • ISBN-13: 9781470456344
Kitos knygos pagal šią temą:
Pastovi nuoroda: https://www.kriso.lt/db/9781470456344.html
Raktažodžiai:
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.
Preface vii
Chapter 1 Introduction
1(8)
1.1 What are Davenport-Zannier polynomials
1(3)
1.2 Dessins d'enfants and Galois theory
4(1)
1.3 What are weighted trees
5(4)
Chapter 2 Dessins d'enfants: from polynomials through Belyi functions to weighted trees
9(10)
2.1 Rational function ƒ = P/R and its critical values
9(2)
2.2 Dessins d'enfants and Belyi functions
11(3)
2.3 Number fields
14(2)
2.4 Additional remarks about weighted trees
16(3)
Chapter 3 Existence theorem
19(8)
3.1 Readability of coverings
19(1)
3.2 Forests
20(1)
3.3 Stitching several trees to get one: the case gcd(α, β) = 1
20(2)
3.4 Non-coprime weights
22(1)
3.5 Weak bound: polynomials and cacti
22(3)
3.6 Proof of the weak bound
25(2)
Chapter 4 Recapitulation and perspectives
27(2)
Chapter 5 Classification of unitrees
29(30)
5.1 Statement of the main result
29(7)
5.2 Weight distribution
36(3)
5.3 Brushes
39(2)
5.4 Trees with repeating branches of height 2
41(4)
5.5 Trees with repeating branches of the type (1, s, s + 1)
45(5)
5.6 Trees with repeating branches of the type (1, t, 1)
50(3)
5.7 The trees A, B, ..., T listed in Theorem 5.4 are unitrees
53(3)
5.8 Appendix: the inverse enumeration problem
56(3)
Chapter 6 Computation of Davenport--Zannier pairs for unitrees
59(28)
6.1 Reciprocal polynomials
59(1)
6.2 Remarks about computation
60(1)
6.3 Stars and binomial series
61(1)
6.4 Forks and Hall's conjecture
62(2)
6.5 Jacobi polynomials
64(8)
6.6 Series F and G: trees of diameter 4
72(5)
6.7 Series H and J: decomposable ordinary trees
77(3)
6.8 Series J
80(2)
6.9 Sporadic trees
82(5)
Chapter 7 Primitive monodromy groups of weighted trees
87(12)
7.1 Monodromy group of a dessin
87(2)
7.2 Primitive and imprimitive groups
89(2)
7.3 A menagerie of groups
91(1)
7.4 Primitive groups containing a cycle: Gareth Jones's classification
92(2)
7.5 Main theorem: statement and commentaries
94(1)
7.6 Main theorem: proof
95(4)
Chapter 8 Trees with primitive monodromy groups
99(38)
Chapter 9 A zoo of examples and constructions
137(18)
9.1 Composition
137(3)
9.2 Difference of powers over Q: infinite series
140(1)
9.3 Polynomials with a relaxed minimum degree condition
141(2)
9.4 Duality and self-duality
143(3)
9.5 A "historic" sporadic example
146(2)
9.6 Some sporadic examples of Beukers and Stewart
148(4)
9.7 Sporadic examples of "megamap invariant"
152(2)
9.8 One more application due to David Roberts
154(1)
Chapter 10 Diophantine invariants
155(8)
10.1 Pell's equation: preliminaries
155(3)
10.2 When a quadratic orbit splits into two rational ones
158(2)
10.3 When the discriminant is a perfect square
160(1)
10.4 Numerical data
160(3)
Chapter 11 Enumeration
163(12)
11.1 Enumeration according to the weight and the number of edges
163(2)
11.2 Dyck words and weighted Dyck words
165(2)
11.3 Proof of the three enumerative theorems
167(2)
11.4 Enumeration according to a passport
169(6)
Chapter 12 What remains to be done
175(4)
Bibliography 179(6)
Index 185
Nikolai M. Adrianov, Lomonosov Moscow State University, Russia

Fedor Pakovich, Ben Gurion University of the Negev, Be'er Sheva, Israel

Alexander K. Zvonkin, University of Bordeaux, Talence, France