Effective Results and Methods for Diophantine Equations over Finitely Generated Domains

Description

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. Cover Series page Title page Copyright page Dedication Contents Preface Acknowledgments Glossary of Frequently Used Notation History and Summary 1 Ineffective Results for Diophantine Equations over Finitely Generated Domains 1.1 Thue Equations 1.2 Unit Equations in Two Unknowns 1.3 Hyper- and Superelliptic Equations 1.4 Curves with Finitely Many Integral Points 1.5 Decomposable Form Equations and Multivariate Unit Equations 1.6 Discriminant Equations for Polynomials and Integral Elements 2 Effective Results for Diophantine Equations over Finitely Generated Domains: The Statements 2.1 Notation and Preliminaries 2.2 Unit Equations in Two Unknowns 2.3 Thue Equations 2.4 Hyper- and Superelliptic Equations, the Schinzel–Tijdeman Equation 2.5 The Catalan Equation 2.6 Decomposable Form Equations 2.7 Norm Form Equations 2.8 Discriminant Form Equations and Discriminant Equations 2.9 Open Problems 3 A Brief Explanation of Our Effective Methods over Finitely Generated Domains 3.1 Sketch of the Effective Specialization Method 3.2 Illustration of the Application of the Effective Specialization Method to Diophantine Equations 3.3 Sketch of the Method Reducing Equations to Unit Equations 3.3.1 Effective Finiteness Result for Systems of Unit Equations 3.3.2 Reduction of Decomposable Form Equations to Unit Equations 3.3.3 Quantitative Versions 3.3.4 Reduction of Discriminant Equations to Unit Equations 3.4 Comparison of Our Two Effective Methods 4 Effective Results over Number Fields 4.1 Notation and Preliminaries 4.2 Effective Estimates for Linear Forms in Logarithms 4.3 S-Unit Equations 4.4 Thue Equations 4.5 Hyper- and Superelliptic Equations, the Schinzel–Tijdeman Equation 4.6 The Catalan Equation 4.7 Decomposable Form Equations 4.8 Discriminant Equations 5 Effective Results over Function Fields 5.1 Notation and Preliminaries 5.2 S-Unit Equations 5.3 The Catalan Equation 5.4 Thue Equations 5.5 Hyper- and Superelliptic Equations 6 Tools from Effective Commutative Algebra 6.1 Effective Linear Algebra over Polynomial Rings 6.2 Finitely Generated Fields over Q 6.3 Finitely Generated Integral Domains over Z 7 The Effective Specialization Method 7.1 Notation 7.2 Construction of a More Convenient Ground Domain B 7.3 Comparison of Different Degrees and Heights 7.4 Specializations 7.5 Multiplicative Independence 8 Degree-Height Estimates 8.1 Definitions 8.2 Estimates for Factors of Polynomials 8.3 Consequences 9 Proofs of the Results from Sections 2.2 to 2.5 Use of Specializations 9.1 A Reduction 9.1.1 Unit Equations 9.1.2 Thue Equations 9.1.3 Hyper- and Superelliptic Equations 9.2 Bounding the Degrees 9.2.1 Unit Equations 9.2.2 Thue Equations 9.2.3 Hyper- and Superelliptic Equations 9.3 Bounding the Heights and Specializations 9.3.1 Unit Equations 9.3.2 Thue Equations 9.3.3 Hyper- and Superelliptic Equations 9.4 The Catalan Equation 10 Proofs of the Results from Sections 2.6 to 2.8 Reduction to Unit Equations 10.1 Proofs of the Central Results on Decomposable Form Equations 10.2 Proofs of the Results for Norm Form Equations 10.3 Proofs of the Results for Discriminant Form Equations and Discriminant Equations References Index