Lecture Notes on Diophantine Analysis
These lecture notes originate from a course delivered at the Scuola Normale in Pisa in 2006. Generally speaking, the prerequisites do not go beyond basic mathematical material and are accessible to many undergraduates. The contents mainly concern diophant
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Lecture Notes on Diophantine Analysis with an appendix by Francesco Amoroso
EDIZIONI DELLA NORMALE
8
APPUNTI LECTURE NOTES
Umberto Zannier Scuola Normale Superiore Piazza dei Cavalieri, 7 56126 Pisa, Italy Lecture Notes on Diophantine Analysis
Umberto Zannier
Lecture Notes on Diophantine Analysis with an appendix by Francesco Amoroso
c 2014 Scuola Normale Superiore Pisa Versione rivista e aggiornata Prima edizione: 2009 ISBN 978-88-7642-341-3 ISBN 978-88-7642-517-2 (eBook)
Contents
Preface
ix
Introduction
xiii
1 Some classical diophantine examples 1.1 The case of a single variable . . . . . . . . . . . . . . . 1.2 The linear case in two variables . . . . . . . . . . . . . . 1.3 Diophantine Approximation . . . . . . . . . . . . . . . 1.4 Pell Equation . . . . . . . . . . . . . . . . . . . . . . . 1.4.1 Structure of the solutions and units in quadratic Welds . . . . . . . . . . . . . . . . . . . . . . . 1.4.2 Effective solution of Pell and related equations . 1.5 The general case of degree 2 . . . . . . . . . . . . . . . Supplements . . . . . . . . . . . . . . . . . . . . . . . . . . . Two applications of Dirichlet Lemma . . . . . . . . . . . A cyclotomic solution of certain Pell equations . . . . . . A Pell Equation in polynomials . . . . . . . . . . . . . . Pad´e Approximations to exp(x) and celebrated irrationalities . . . . . . . . . . . . . . . . . . . . . . . Rational points on conics . . . . . . . . . . . . . . . . . A theorem of Fermat . . . . . . . . . . . . . . . . . . . . Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
28 30 31 32
2 Thue’s equations and rational approximations 2.1 Thue Equations . . . . . . . . . . . . . . . . . . . 2.2 Rational approximations to algebraic numbers . . . 2.3 Thue’s method and later developements . . . . . . 2.3.1 A rough sketch of Thue’s proof . . . . . . 2.3.2 A reformulation and some later reWnements 2.4 Proof of Thue’s Approximation Theorem . . . . .
37 37 42 46 46 48 51
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1 1 2 4 8 11 14 19 22 23 25 26
vi Umberto Zannier
2.4.1 Preliminaries . . . . . . . . . . . . . . . . . . . 2.4.2 Construction of polynomials F n . . . . . . . . . 2.4.3 Upper bound for D j Fn (u, v) . . . . . . . . . . 2.4.4 Lower bound for |Di Fn (u, v)|. . . . . . . . . . . 2.4.5 An upper bound for the multiplicity at (u, v) . . 2.4.6 Conclusions . . . . . . . . . . . . . . . . . . . . Supplements . . . . . . . . . . . . . . . . . . . . . . . . . . . Finiteness of integral points on certain curves . . . . . . . Effective decision for an inWnity of integral points in genus zero . . . . . . . . . . . . . . . . . . . . . . . . . . . . A theorem of Runge . . . . . . . . . . . . . . . . . . . . A Thue Equation in polynomials . . . . . . . . . . . . . Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
51 53 56 57 58 60 63 63 69 69 73 74
3 Heights and diophantine equations over number Helds 3.1 Fields with a product formula . . . . . . . . . . . . . . 3.1.1 Valuations and the product formula . . . . . .
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