Search

Effective Results and Methods for Diophantine Equations over Finitely Generated Domains

Jan-Hendrik Evertse, Kálmán Győry
Published online:

31 August 2022

Print publication:

28 April 2022
- Book
- - Get access
    
    Buy a print copy
    
    Check if you have access via personal or institutional login
    
    Log in Register
- Export citation
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.

1 - Ineffective Results for Diophantine Equations over Finitely Generated Domains
Jan-Hendrik Evertse, Universiteit Leiden, Kálmán Győry, Debreceni Egyetem, Hungary
Book:

Effective Results and Methods for Diophantine Equations over Finitely Generated Domains

Published online:

31 August 2022

Print publication:

28 April 2022, pp 1-17
- Chapter
- - Get access
    
    Check if you have access via personal or institutional login
    
    Log in Register
- Export citation
Summary

We give an overview of the history of the classical Diophantine equations treated in our book and describe the ineffective finiteness results for those. We consider unit equations, Thue equations, hyper- and superelliptic equations, the Schinzel–Tijdeman equation, Catalan’s equation, decomposable form equations, and discriminant equations.

References
Jan-Hendrik Evertse, Universiteit Leiden, Kálmán Győry, Debreceni Egyetem, Hungary
Book:

Effective Results and Methods for Diophantine Equations over Finitely Generated Domains

Published online:

31 August 2022

Print publication:

28 April 2022, pp 206-213
- Chapter
- - Get access
    
    Check if you have access via personal or institutional login
    
    Log in Register
- Export citation

2 - Effective Results for Diophantine Equations over Finitely Generated Domains: The Statements
Jan-Hendrik Evertse, Universiteit Leiden, Kálmán Győry, Debreceni Egyetem, Hungary
Book:

Effective Results and Methods for Diophantine Equations over Finitely Generated Domains

Published online:

31 August 2022

Print publication:

28 April 2022, pp 18-38
- Chapter
- - Get access
    
    Check if you have access via personal or institutional login
    
    Log in Register
- Export citation
Summary

We give the necessary definitions concerning finitely generated domains and effectiveness, and formulate our theorems.

History and Summary
Jan-Hendrik Evertse, Universiteit Leiden, Kálmán Győry, Debreceni Egyetem, Hungary
Book:

Effective Results and Methods for Diophantine Equations over Finitely Generated Domains

Published online:

31 August 2022

Print publication:

28 April 2022, pp xix-xxiv
- Chapter
- - Get access
    
    Check if you have access via personal or institutional login
    
    Log in Register
- Export citation

3 - A Brief Explanation of Our Effective Methods over Finitely Generated Domains
Jan-Hendrik Evertse, Universiteit Leiden, Kálmán Győry, Debreceni Egyetem, Hungary
Book:

Effective Results and Methods for Diophantine Equations over Finitely Generated Domains

Published online:

31 August 2022

Print publication:

28 April 2022, pp 39-54
- Chapter
- - Get access
    
    Check if you have access via personal or institutional login
    
    Log in Register
- Export citation
Summary

We outline the methods used in our proofs, especially our specialization techniques and reduction to unit equations.

Dedication
Jan-Hendrik Evertse, Universiteit Leiden, Kálmán Győry, Debreceni Egyetem, Hungary
Book:

Effective Results and Methods for Diophantine Equations over Finitely Generated Domains

Published online:

31 August 2022

Print publication:

28 April 2022, pp v-vi
- Chapter
- - Get access
    
    Check if you have access via personal or institutional login
    
    Log in Register
- Export citation

10 - Proofs of the Results from Sections 2.6 to 2.8 Reduction to Unit Equations
Jan-Hendrik Evertse, Universiteit Leiden, Kálmán Győry, Debreceni Egyetem, Hungary
Book:

Effective Results and Methods for Diophantine Equations over Finitely Generated Domains

Published online:

31 August 2022

Print publication:

28 April 2022, pp 194-205
- Chapter
- - Get access
    
    Check if you have access via personal or institutional login
    
    Log in Register
- Export citation
Summary

We complete the proofs of our results for decomposable form equations and discriminant equations, by making a reduction to the effective finiteness results for unit equations proved in Chapter 9. In the course of the proof, we apply the estimates for degree-height estimates, proved in Chapter 8.

6 - Tools from Effective Commutative Algebra
Jan-Hendrik Evertse, Universiteit Leiden, Kálmán Győry, Debreceni Egyetem, Hungary
Book:

Effective Results and Methods for Diophantine Equations over Finitely Generated Domains

Published online:

31 August 2022

Print publication:

28 April 2022, pp 114-127
- Chapter
- - Get access
    
    Check if you have access via personal or institutional login
    
    Log in Register
- Export citation
Summary

We give an overview of the techniques from commutative algebra that we need. In particular, we recall explicit results of Aschenbrenner concerning the solutions of linear equations over polynomial rings over the integers.

Contents
Jan-Hendrik Evertse, Universiteit Leiden, Kálmán Győry, Debreceni Egyetem, Hungary
Book:

Effective Results and Methods for Diophantine Equations over Finitely Generated Domains

Published online:

31 August 2022

Print publication:

28 April 2022, pp vii-x
- Chapter
- - Get access
    
    Check if you have access via personal or institutional login
    
    Log in Register
- Export citation

9 - Proofs of the Results from Sections 2.2 to 2.5 Use of Specializations
Jan-Hendrik Evertse, Universiteit Leiden, Kálmán Győry, Debreceni Egyetem, Hungary
Book:

Effective Results and Methods for Diophantine Equations over Finitely Generated Domains

Published online:

31 August 2022

Print publication:

28 April 2022, pp 171-193
- Chapter
- - Get access
    
    Check if you have access via personal or institutional login
    
    Log in Register
- Export citation
Summary

We complete the proofs of our effective finiteness results for unit equations, Thue equations, hyper- and superelliptic equations, the Schinzel–Tijdeman equation, and Catalan’s equation. We apply the specialization method from Chapter 7 and make a reduction to the results from Chapters 4 and 5.

Glossary of Frequently Used Notation
Jan-Hendrik Evertse, Universiteit Leiden, Kálmán Győry, Debreceni Egyetem, Hungary
Book:

Effective Results and Methods for Diophantine Equations over Finitely Generated Domains

Published online:

31 August 2022

Print publication:

28 April 2022, pp xv-xviii
- Chapter
- - Get access
    
    Check if you have access via personal or institutional login
    
    Log in Register
- Export citation

8 - Degree-Height Estimates
Jan-Hendrik Evertse, Universiteit Leiden, Kálmán Győry, Debreceni Egyetem, Hungary
Book:

Effective Results and Methods for Diophantine Equations over Finitely Generated Domains

Published online:

31 August 2022

Print publication:

28 April 2022, pp 156-170
- Chapter
- - Get access
    
    Check if you have access via personal or institutional login
    
    Log in Register
- Export citation
Summary

we introduce degree-height estimates for elements of finitely generated domains and of algebraic extensions thereof, which may be seen as an analogue of the naive height of an algebraic number, and prove some properties of those.

7 - The Effective Specialization Method
Jan-Hendrik Evertse, Universiteit Leiden, Kálmán Győry, Debreceni Egyetem, Hungary
Book:

Effective Results and Methods for Diophantine Equations over Finitely Generated Domains

Published online:

31 August 2022

Print publication:

28 April 2022, pp 128-155
- Chapter
- - Get access
    
    Check if you have access via personal or institutional login
    
    Log in Register
- Export citation
Summary

We describe in full detail our effective specialization method, which allows us to reduce effective finiteness results for Diophantine equations over finitely generated domains to effective results over number fields and function fields, discussed in Chapters 4 and 5.

Preface
Jan-Hendrik Evertse, Universiteit Leiden, Kálmán Győry, Debreceni Egyetem, Hungary
Book:

Effective Results and Methods for Diophantine Equations over Finitely Generated Domains

Published online:

31 August 2022

Print publication:

28 April 2022, pp xi-xiii
- Chapter
- - Get access
    
    Check if you have access via personal or institutional login
    
    Log in Register
- Export citation

Acknowledgments
Jan-Hendrik Evertse, Universiteit Leiden, Kálmán Győry, Debreceni Egyetem, Hungary
Book:

Effective Results and Methods for Diophantine Equations over Finitely Generated Domains

Published online:

31 August 2022

Print publication:

28 April 2022, pp xiv-xiv
- Chapter
- - Get access
    
    Check if you have access via personal or institutional login
    
    Log in Register
- Export citation

Frontmatter
Jan-Hendrik Evertse, Universiteit Leiden, Kálmán Győry, Debreceni Egyetem, Hungary
Book:

Effective Results and Methods for Diophantine Equations over Finitely Generated Domains

Published online:

31 August 2022

Print publication:

28 April 2022, pp i-iv
- Chapter
- - Get access
    
    Check if you have access via personal or institutional login
    
    Log in Register
- Export citation

Index
Jan-Hendrik Evertse, Universiteit Leiden, Kálmán Győry, Debreceni Egyetem, Hungary
Book:

Effective Results and Methods for Diophantine Equations over Finitely Generated Domains

Published online:

31 August 2022

Print publication:

28 April 2022, pp 214-216
- Chapter
- - Get access
    
    Check if you have access via personal or institutional login
    
    Log in Register
- Export citation

5 - Effective Results over Function Fields
Jan-Hendrik Evertse, Universiteit Leiden, Kálmán Győry, Debreceni Egyetem, Hungary
Book:

Effective Results and Methods for Diophantine Equations over Finitely Generated Domains

Published online:

31 August 2022

Print publication:

28 April 2022, pp 98-113
- Chapter
- - Get access
    
    Check if you have access via personal or institutional login
    
    Log in Register
- Export citation
Summary

We recall the basics for function fields of characteristic 0, state and prove the Mason–Stothers abc-theorem for function fields, and deduce effective finiteness results for various classes of Diophantine equations.

4 - Effective Results over Number Fields
Jan-Hendrik Evertse, Universiteit Leiden, Kálmán Győry, Debreceni Egyetem, Hungary
Book:

Effective Results and Methods for Diophantine Equations over Finitely Generated Domains

Published online:

31 August 2022

Print publication:

28 April 2022, pp 55-97
- Chapter
- - Get access
    
    Check if you have access via personal or institutional login
    
    Log in Register
- Export citation
Summary

We recall Baker type results on logarithmic forms, give an overview of the necessary techniques from algebraic number theory, and sketch the proofs of effective finiteness results for various classes of Diophantine equations.

Search Results

Refine search

Refine search

Actions for selected content:

75 results

Effective Results and Methods for Diophantine Equations over Finitely Generated Domains

1 - Ineffective Results for Diophantine Equations over Finitely Generated Domains

Summary

References

2 - Effective Results for Diophantine Equations over Finitely Generated Domains: The Statements

Summary

History and Summary

3 - A Brief Explanation of Our Effective Methods over Finitely Generated Domains

Summary

Dedication

10 - Proofs of the Results from Sections 2.6 to 2.8 Reduction to Unit Equations

Summary

6 - Tools from Effective Commutative Algebra

Summary

Contents

9 - Proofs of the Results from Sections 2.2 to 2.5 Use of Specializations

Summary

Glossary of Frequently Used Notation

8 - Degree-Height Estimates

Summary

7 - The Effective Specialization Method

Summary

Preface

Acknowledgments

Frontmatter

Index

5 - Effective Results over Function Fields

Summary

4 - Effective Results over Number Fields

Summary

Search Results

Refine search

Refine search

Actions for selected content:

Save Search

75 results

Effective Results and Methods for Diophantine Equations over Finitely Generated Domains

Summary

Summary

Summary

Summary

Summary

Summary

Summary

Summary

Summary

Summary