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El. knyga: Unit Equations in Diophantine Number Theory

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(Universiteit Leiden), (Debreceni Egyetem, Hungary)
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Unit equations play a central role in Diophantine number theory. This book provides a comprehensive and up-to-date treatment of unit equations and their various applications. It brings together the most important results and gives an overview of the basic techniques, making it accessible to young researchers.

Diophantine number theory is an active area that has seen tremendous growth over the past century, and in this theory unit equations play a central role. This comprehensive treatment is the first volume devoted to these equations. The authors gather together all the most important results and look at many different aspects, including effective results on unit equations over number fields, estimates on the number of solutions, analogues for function fields and effective results for unit equations over finitely generated domains. They also present a variety of applications. Introductory chapters provide the necessary background in algebraic number theory and function field theory, as well as an account of the required tools from Diophantine approximation and transcendence theory. This makes the book suitable for young researchers as well as experts who are looking for an up-to-date overview of the field.

Recenzijos

'The book is well written and is certain to be of use to experts and graduate students alike. Every chapter is prefaced with a nice introduction and summary, putting the material in perspective and surveying the proof techniques. The book also features an extensive bibliography and an easy-to-use glossary and index.' Lenny Fukshansky, MathSciNet 'Understanding the book requires only basic knowledge in algebra (groups, commutative rings, fields, Galois theory and elementary algebraic number theory). In particular the concepts of height, places and valuations play an important role. In addition to providing a survey, the authors improve both formulations and proofs of various important existing results in the literature, making their book a valuable asset for researchers in this area. I thank them for their fast and clear answers.' Pieter Moree, Nieuw Archief voor Wiskunde

Daugiau informacijos

A comprehensive, graduate-level treatment of unit equations and their various applications.
Preface ix
Summary xi
PART I PRELIMINARIES
1 Basic algebraic number theory
3(27)
1.1 Characteristic polynomial, trace, norm, discriminant
3(2)
1.2 Ideal theory for algebraic number fields
5(2)
1.3 Extension of ideals; norm of ideals
7(2)
1.4 Discriminant, class number, unit group and regulator
9(2)
1.5 Explicit estimates
11(1)
1.6 Absolute values: generalities
12(3)
1.7 Absolute values and places on number fields
15(2)
1.8 5-integers, S-units and S-norm
17(2)
1.9 Heights
19(4)
1.9.1 Heights of algebraic numbers
19(2)
1.9.2 V-adic norms and heights of vectors and polynomials
21(2)
1.10 Effective computations in number fields
23(3)
1.11 p-adic numbers
26(4)
2 Algebraic function fields
30(12)
2.1 Valuations
30(3)
2.2 Heights
33(2)
2.3 Derivatives and genus
35(2)
2.4 Effective computations
37(5)
3 Tools from Diophantine approximation and transcendence theory
42(19)
3.1 The Subspace Theorem and some variations
42(9)
3.2 Effective estimates for linear forms in logarithms
51(10)
PART II UNIT EQUATIONS AND APPLICATIONS
4 Effective results for unit equations in two unknowns over number fields
61(35)
4.1 Effective bounds for the heights of the solutions
62(5)
4.1.1 Equations in units of a number field
62(2)
4.1.2 Equations with unknowns from a finitely generated multiplicative group
64(3)
4.2 Approximation by elements of a finitely generated multiplicative group
67(1)
4.3 Tools
68(11)
4.3.1 Some geometry of numbers
68(4)
4.3.2 Estimates for units and S-units
72(7)
4.4 Proofs
79(8)
4.4.1 Proofs of Theorems 4.1.1 and 4.1.2
79(2)
4.4.2 Proofs of Theorems 4.2.1 and 4.2.2
81(3)
4.4.3 Proofs of Theorem 4.1.3 and its corollaries
84(3)
4.5 Alternative methods, comparison of the bounds
87(2)
4.5.1 The results of Bombieri, Bombieri and Cohen, and Bugeaud
87(1)
4.5.2 The results of Murty, Pasten and von Kanel
88(1)
4.6 The abc-conjecture
89(4)
4.7 Notes
93(3)
4.7.1 Historical remarks and some related results
93(1)
4.7.2 Some notes on applications
94(2)
5 Algorithmic resolution of unit equations in two unknowns
96(32)
5.1 Application of Baker's type estimates
97(6)
5.1.1 Infinite places
100(2)
5.1.2 Finite places
102(1)
5.2 Reduction of the bounds
103(8)
5.2.1 Infinite places
103(2)
5.2.2 Finite places
105(6)
5.3 Enumeration of the "small" solutions
111(8)
5.4 Examples
119(2)
5.5 Exceptional units
121(2)
5.6 Supplement: LLL lattice basis reduction
123(3)
5.7 Notes
126(2)
6 Unit equations in several unknowns
128(45)
6.1 Results
130(6)
6.1.1 A semi-effective result
130(1)
6.1.2 Upper bounds for the number of solutions
131(3)
6.1.3 Lower bounds
134(2)
6.2 Proofs of Theorem 6.1.1 and Corollary 6.1.2
136(4)
6.3 A sketch of the proof of Theorem 6.1.3
140(8)
6.3.1 A reduction
140(2)
6.3.2 Notation
142(1)
6.3.3 Covering results
142(2)
6.3.4 The large solutions
144(3)
6.3.5 The small solutions, and conclusion of the proof
147(1)
6.4 Proof of Theorem 6.1.4
148(10)
6.5 Proof of Theorem 6.1.6
158(3)
6.6 Proofs of Theorems 6.1.7 and 6.1.8
161(4)
6.7 Notes
165(8)
7 Analogues over function fields
173(24)
7.1 Mason's inequality
174(2)
7.2 Proofs
176(2)
7.3 Effective results in the more unknowns case
178(4)
7.4 Results on the number of solutions
182(1)
7.5 Proof of Theorem 7.4.1
183(9)
7.5.1 Extension to the k-closure of Γ
183(2)
7.5.2 Some algebraic geometry
185(3)
7.5.3 Proof of Theorem 7.5.1
188(4)
7.6 Results in positive characteristic
192(5)
8 Effective results for unit equations in two unknowns over finitely generated domains
197(34)
8.1 Statements of the results
198(3)
8.2 Effective linear algebra over polynomial rings
201(3)
8.3 A reduction
204(8)
8.4 Bounding the degree in Proposition 8.3.7
212(3)
8.5 Specializations
215(7)
8.6 Bounding the height in Proposition 8.3.7
222(3)
8.7 Proof of Theorem 8.1.3
225(5)
8.8 Notes
230(1)
9 Decomposable form equations
231(53)
9.1 A finiteness criterion for decomposable form equations
233(3)
9.2 Reduction of unit equations to decomposable form equations
236(1)
9.3 Reduction of decomposable form equations to unit equations
237(7)
9.3.1 Proof of the equivalence (ii) ↔ (iii) in Theorem 9.1.1
238(1)
9.3.2 Proof of the implication (i) → (iii) in Theorem 9.1.1
238(2)
9.3.3 Proof of the implication (iii) → (i) in Theorem 9.1.1
240(4)
9.4 Finiteness of the number of families of solutions
244(5)
9.5 Upper bounds for the number of solutions
249(8)
9.5.1 Galois symmetric S-unit vectors
251(2)
9.5.2 Consequences for decomposable form equations and S-unit equations
253(4)
9.6 Effective results
257(15)
9.6.1 Thue equations
258(5)
9.6.2 Decomposable form equations in an arbitrary number of unknowns
263(9)
9.7 Notes
272(12)
10 Further applications
284(53)
10.1 Prime factors of sums of integers
284(3)
10.2 Additive unit representations in finitely generated integral domains
287(4)
10.3 Orbits of polynomial and rational maps
291(7)
10.4 Polynomials dividing many k-nomials
298(3)
10.5 Irreducible polynomials and arithmetic graphs
301(4)
10.6 Discriminant equations and power integral bases in number fields
305(5)
10.7 Binary forms of given discriminant
310(5)
10.8 Resultant equations for monic polynomials
315(2)
10.9 Resultant inequalities and equations for binary forms
317(4)
10.10 Lang's Conjecture for tori
321(5)
10.11 Linear recurrence sequences and exponential-polynomial equations
326(4)
10.12 Algebraic independence results
330(7)
References 337(21)
Glossary of frequently used notation 358(3)
Index 361
Jan-Hendrik Evertse is Assistant Professor in the Mathematical Institute at Leiden University. His research concentrates on Diophantine approximation and applications to Diophantine problems. In this area he has obtained some influential results, in particular on estimates for the numbers of solutions of Diophantine equations and inequalities. He has written more than 75 research papers and co-authored one book with Bas Edixhoven entitled Diophantine Approximation and Abelian Varieties (2003). Kįlmįn Gyry is Professor Emeritus at the University of Debrecen, a member of the Hungarian Academy of Sciences and a well-known researcher in Diophantine number theory. Over his career he has obtained several significant and pioneering results, among others on unit equations, decomposable form equations, and their various applications. His results have been published in one book and 160 research papers. Gyry is also the founder and leader of the Number Theory Research Group in Debrecen, which consists of his former students and their descendants.
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