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Discriminant Equations in Diophantine Number Theory

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Part of New Mathematical Monographs

  • Date Published: November 2016
  • availability: Available
  • format: Hardback
  • isbn: 9781107097612

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About the Authors
  • Discriminant equations are an important class of Diophantine equations with close ties to algebraic number theory, Diophantine approximation and Diophantine geometry. This book is the first comprehensive account of discriminant equations and their applications. It brings together many aspects, including effective results over number fields, effective results over finitely generated domains, estimates on the number of solutions, applications to algebraic integers of given discriminant, power integral bases, canonical number systems, root separation of polynomials and reduction of hyperelliptic curves. The authors' previous title, Unit Equations in Diophantine Number Theory, laid the groundwork by presenting important results that are used as tools in the present book. This material is briefly summarized in the introductory chapters along with the necessary basic algebra and algebraic number theory, making the book accessible to experts and young researchers alike.

    • Gathers important results on discriminant equations and makes them accessible to experts and young researchers alike
    • Considers many different aspects that may stimulate further research in the area
    • The authors draw on their 40 years of experience in the field
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    Reviews & endorsements

    '… the book is very interesting and well written. It contains the motivational material necessary for those entering in the field of discriminant equations and succeeds to bring the reader to the forefront of research. Graduates and researchers in the field of number theory will find it a very valuable resource.' Dimitros Poulakis, Zentralblatt MATH

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    Product details

    • Date Published: November 2016
    • format: Hardback
    • isbn: 9781107097612
    • length: 476 pages
    • dimensions: 236 x 158 x 35 mm
    • weight: 0.86kg
    • availability: Available
  • Table of Contents

    Preface
    Summary
    Part I. Preliminaries:
    1. Finite étale algebras over fields
    2. Dedekind domains
    3. Algebraic number fields
    4. Tools from the theory of unit equations
    Part II. Monic Polynomials and Integral Elements of Given Discriminant, Monogenic Orders:
    5. Basic finiteness theorems
    6. Effective results over Z
    7. Algorithmic resolution of discriminant form and index form equations
    8. Effective results over the S-integers of a number field
    9. The number of solutions of discriminant equations
    10. Effective results over finitely generated domains
    11. Further applications
    Part III. Binary Forms of Given Discriminant:
    12. A brief overview of the basic finiteness theorems
    13. Reduction theory of binary forms
    14. Effective results for binary forms of given discriminant
    15. Semi-effective results for binary forms of given discriminant
    16. Invariant orders of binary forms
    17. On the number of equivalence classes of binary forms of given discriminant
    18. Further applications
    Glossary of frequently used notation
    References
    Index.

  • Authors

    Jan-Hendrik Evertse, Universiteit Leiden
    Jan-Hendrik Evertse works 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.

    Kálmán Győry, Debreceni Egyetem, Hungary
    Kálmán Győry 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 and decomposable form equations, and their various applications. Győry 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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