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Effective Results and Methods for Diophantine Equations over Finitely Generated Domains [Pehme köide]

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(Universiteit Leiden), (Debreceni Egyetem, Hungary)
  • Formaat: Paperback / softback, 240 pages, kõrgus x laius x paksus: 230x152x13 mm, kaal: 360 g, Worked examples or Exercises
  • Sari: London Mathematical Society Lecture Note Series
  • Ilmumisaeg: 28-Apr-2022
  • Kirjastus: Cambridge University Press
  • ISBN-10: 1009005855
  • ISBN-13: 9781009005852
Teised raamatud teemal:
Effective Results and Methods for Diophantine Equations over Finitely Generated DomainsSuurem pilt
  • Formaat: Paperback / softback, 240 pages, kõrgus x laius x paksus: 230x152x13 mm, kaal: 360 g, Worked examples or Exercises
  • Sari: London Mathematical Society Lecture Note Series
  • Ilmumisaeg: 28-Apr-2022
  • Kirjastus: Cambridge University Press
  • ISBN-10: 1009005855
  • ISBN-13: 9781009005852
Teised raamatud teemal:
Püsilink: https://www.kriso.ee/db/9781009005852.html
Märksõnad:
This book provides the first thorough treatment of effective results and methods for Diophantine equations over finitely generated domains. Compiling diverse results and techniques from papers written in recent decades, the text includes an in-depth analysis of classical equations including unit equations, Thue equations, hyper- and superelliptic equations, the Catalan equation, discriminant equations and decomposable form equations. The majority of results are proved in a quantitative form, giving effective bounds on the sizes of the solutions. The necessary techniques from Diophantine approximation and commutative algebra are all explained in detail without requiring any specialized knowledge on the topic, enabling readers from beginning graduate students to experts to prove effective finiteness results for various further classes of Diophantine equations.

Arvustused

' I found the book to be presented and structured very well. It covers the topics and results that one would expect and hope to find in a book on this subject, as well as the new results mentioned above. But as the authors state towards the end of their preface, more possibilities exist for the application of their techniques. The authors have certainly done a good job of writing a clear, accessible account of this subject that should help to fulfill their hope that others will continue their work.' Paul M. Voutier, MathSciNet

Muu info

Provides exceptional coverage of effective solutions for Diophantine equations over finitely generated domains.
Preface xi
Acknowledgments xiv
Glossary of Frequently Used Notation xv
History and Summary xix
1 Ineffective Results for Diophantine Equations over Finitely Generated Domains
1(17)
1.1 Thue Equations
2(3)
1.2 Unit Equations in Two Unknowns
5(2)
1.3 Hyper- and Superelliptic Equations
7(1)
1.4 Curves with Finitely Many Integral Points
8(1)
1.5 Decomposable Form Equations and Multivariate Unit Equations
9(4)
1.6 Discriminant Equations for Polynomials and Integral Elements
13(5)
2 Effective Results for Diophantine Equations over Finitely Generated Domains: The Statements
18(21)
2.1 Notation and Preliminaries
18(3)
2.2 Unit Equations in Two Unknowns
21(3)
2.3 Thue Equations
24(1)
2.4 Hyper- and Superelliptic Equations, the Schinzel-Tijdeman Equation
24(1)
2.5 The Catalan Equation
25(1)
2.6 Decomposable Form Equations
26(5)
2.7 Norm Form Equations
31(1)
2.8 Discriminant Form Equations and Discriminant Equations
32(4)
2.9 Open Problems
36(3)
3 A Brief Explanation of Our Effective Methods over Finitely Generated Domains
39(16)
3.1 Sketch of the Effective Specialization Method
39(6)
3.2 Illustration of the Application of the Effective Specialization Method to Diophantine Equations
45(1)
3.3 Sketch of the Method Reducing Equations to Unit Equations
46(8)
3.3.1 Effective Finiteness Result for Systems of Unit Equations
47(2)
3.3.2 Reduction of Decomposable Form Equations to Unit Equations
49(1)
3.3.3 Quantitative Versions
50(2)
3.3.4 Reduction of Discriminant Equations to Unit Equations
52(2)
3.4 Comparison of Our Two Effective Methods
54(1)
4 Effective Results over Number Fields
55(43)
4.1 Notation and Preliminaries
56(8)
4.2 Effective Estimates for Linear Forms in Logarithms
64(3)
4.3 5-Unit Equations
67(4)
4.4 Thue Equations
71(2)
4.5 Hyper- and Superelliptic Equations, the Schinzel--Tijdeman Equation
73(8)
4.6 The Catalan Equation
81(8)
4.7 Decomposable Form Equations
89(5)
4.8 Discriminant Equations
94(4)
5 Effective Results over Function Fields
98(16)
5.1 Notation and Preliminaries
98(4)
5.2 S-Unit Equations
102(2)
5.3 The Catalan Equation
104(1)
5.4 Thue Equations
105(3)
5.5 Hyper-and Superelliptic Equations
108(6)
6 Tools from Effective Commutative Algebra
114(14)
6.1 Effective Linear Algebra over Polynomial Rings
115(4)
6.2 Finitely Generated Fields over Q
119(3)
6.3 Finitely Generated Integral Domains over Z
122(6)
7 The Effective Specialization Method
128(28)
7.1 Notation
128(1)
7.2 Construction of a More Convenient Ground Domain B
129(7)
7.3 Comparison of Different Degrees and Heights
136(4)
7.4 Specializations
140(10)
7.5 Multiplicative Independence
150(6)
8 Degree-Height Estimates
156(15)
8.1 Definitions
156(2)
8.2 Estimates for Factors of Polynomials
158(4)
8.3 Consequences
162(9)
9 Proofs of the Results from Sections 2.2 to 2.5 Use of Specializations
171(23)
9.1 A Reduction
172(5)
9.1.1 Unit Equations
173(2)
9.1.2 Thue Equations
175(1)
9.1.3 Hyper-and Superelliptic Equations
176(1)
9.2 Bounding the Degrees
177(4)
9.2.1 Unit Equations
178(1)
9.2.2 Thue Equations
179(1)
9.2.3 Hyper-and Superelliptic Equations
180(1)
9.3 Bounding the Heights and Specializations
181(9)
9.3.1 Unit Equations
182(2)
9.3.2 Thue Equations
184(4)
9.3.3 Hyper-and Superelliptic Equations
188(2)
9.4 The Catalan Equation
190(4)
10 Proofs of the Results from Sections 2.6 to 2.8 Reduction to Unit Equations
194(12)
10.1 Proofs of the Central Results on Decomposable Form Equations
194(7)
10.2 Proofs of the Results for Norm Form Equations
201(1)
10.3 Proofs of the Results for Discriminant Form Equations and Discriminant Equations
202(4)
References 206(8)
Index 214
Jan-Hendrik Evertse is Associate Professor in Number Theory at Leiden University in the Netherlands. He co-edited the lecture notes in mathematics Diophantine Approximation and Abelian Varieties (1993) with Bas Edixhoven, and co-authored two books with Kálmán Gyry: Unit Equations in Diophantine Number Theory (Cambridge, 2016) and Discriminant Equations in Diophantine Number Theory (Cambridge, 2016). Kálmán Gyry is Professor Emeritus at the University of Debrecen, Hungary and a member of the Hungarian Academy of Sciences. Gyry is the founder and leader of the Number Theory Research Group in Debrecen. Together with Jan-Hendrik Evertse he has written two books: Unit Equations in Diophantine Number Theory (Cambridge, 2016) and Discriminant Equations in Diophantine Number Theory (Cambridge, 2016).