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Auxiliary Polynomials in Number Theory

$140.00 (C)

Part of Cambridge Tracts in Mathematics

  • Date Published: July 2016
  • availability: Available
  • format: Hardback
  • isbn: 9781107061576

$ 140.00 (C)
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About the Authors
  • This unified account of various aspects of a powerful classical method, easy to understand in its simplest forms, is illustrated by applications in several areas of number theory. As well as including diophantine approximation and transcendence, which were mainly responsible for its invention, the author places the method in a broader context by exploring its application in other areas, such as exponential sums and counting problems in both finite fields and the field of rationals. Throughout the book, the method is explained in a 'molecular' fashion, where key ideas are introduced independently. Each application is the most elementary significant example of its kind and appears with detailed references to subsequent developments, making it accessible to advanced undergraduates as well as postgraduate students in number theory or related areas. It provides over 700 exercises both guiding and challenging, while the broad array of applications should interest professionals in fields from number theory to algebraic geometry.

    • The method is placed in a broad context through the inclusion of applications outside diophantine approximation and transcendence
    • Key ideas are introduced independently, along with the motivation for each one
    • Includes over 700 exercises ranging from simple to challenging
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    Reviews & endorsements

    'Several features of this book are original. First of all: the topic … Next, thanks to the unique style of the author, this book offers a pleasant reading; a number of nice jokes enable the reader to have a good time while learning high level serious mathematic … This book includes a large number of statements, proofs, ideas, problem which will be of great value for the specialists; but it should interest also any mathematician, including students, who wish to expand their knowledge and see a superb example of a topic having an surprising number of different applications in several directions.' Bulletin of the European Mathematical Society

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    Product details

    • Date Published: July 2016
    • format: Hardback
    • isbn: 9781107061576
    • length: 368 pages
    • dimensions: 236 x 158 x 28 mm
    • weight: 0.7kg
    • contains: 700 exercises
    • availability: Available
  • Table of Contents

    Introduction
    1. Prologue
    2. Irrationality I
    3. Irrationality II - Mahler's method
    4. Diophantine equations - Runge's method
    5. Irreducibility
    6. Elliptic curves - Stepanov's method
    7. Exponential sums
    8. Irrationality measures I - Mahler
    9. Integer-valued entire functions I - Pólya
    10. Integer-valued entire functions II - Gramain
    11. Transcendence I - Mahler
    12. Irrationality measures II - Thue
    13. Transcendence II - Hermite–Lindemann
    14. Heights
    15. Equidistribution - Bilu
    16. Height lower bounds - Dobrowolski
    17. Height upper bounds
    18. Counting - Bombieri–Pila
    19. Transcendence III - Gelfond–Schneider–Lang
    20. Elliptic functions
    21. Modular functions
    22. Algebraic independence
    Appendix: Néron's square root
    References
    Index.

  • Author

    David Masser, Universität Basel, Switzerland
    David Masser is Emeritus Professor in the Department of Mathematics and Computer Science at the University of Basel, Switzerland. He started his career with Alan Baker, which gave him a grounding in modern transcendence theory and began his fascination with the method of auxiliary polynomials. His subsequent interest in applying the method to areas outside transcendence, which involved mainly problems of zero estimates, culminated in his works with Gisbert Wüstholz on isogeny and polarization estimates for abelian varieties, for which he was elected a Fellow of the Royal Society in 2005. This expertise proved beneficial in his more recent works with Umberto Zannier on problems of unlikely intersections, where zero estimates make a return appearance.

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