Diophantine Equations Over Function Fields

London Mathematical Society Lecture Note Series : 96 Diophantine Equations over Function Fields R.C. MASON Fellow of Gonville and Caius College , Cambridge CAMBRIDGE UNIVERSITY PRESS Cambridge London New York New Rochelle Melbourne ...

Author: R. C. Mason

Publisher: Cambridge University Press

ISBN: 0521269830

Category: Mathematics

Page: 142

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A self-contained account of a new approach to the subject.

Effective Results and Methods for Diophantine Equations over Finitely Generated Domains

Soc. 24, 414–426. Mason R. C. (1983), The hyperelliptic equation over function fields, Math. Proc. Camb. Phil. Soc. 93, 219–230. Mason R. C. (1984), Diophantine Equations over Function Fields, Cambridge University Press.

Author: Jan-Hendrik Evertse

Publisher: Cambridge University Press

ISBN: 9781009050036

Category: Mathematics

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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.

Unit Equations in Diophantine Number Theory

Ga ́al, I. (1988b), Inhomogeneous norm form equations over function fields, Acta Arith. 51, 61–73. Ga ́al, I. (2002), Diophantine equations and power integral bases, Birkh ̈auser. Ga ́al, I. and M. Pohst (2002), On the resolution of ...

Author: Jan-Hendrik Evertse

Publisher: Cambridge University Press

ISBN: 9781107097605

Category: Mathematics

Page: 381

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A comprehensive, graduate-level treatment of unit equations and their various applications.

On Finiteness in Differential Equations and Diophantine Geometry

References [ BM ] D. Brownawell and D. Masser , Vanishing sums in function fields , Math . Proc . Cambridge Philos . Soc . 100 ( 1986 ) , 427-434 . [ B1 ] A. Buium , Fields of definition of algebraic varieties in characteristic zero ...

Author: Dana Schlomiuk

Publisher: American Mathematical Soc.

ISBN: 082186985X

Category: Mathematics

Page: 200

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This book focuses on finiteness conjectures and results in ordinary differential equations (ODEs) and Diophantine geometry. During the past twenty-five years, much progress has been achieved on finiteness conjectures, which are the offspring of the second part of Hilbert's 16th problem. Even in its simplest case, this is one of the very few problems on Hilbert's list which remains unsolved. These results are about existence and estimation of finite bounds for the number of limit cycles occurring in certain families of ODEs. The book describes this progress, the methods used (bifurcation theory, asymptotic expansions, methods of differential algebra, or geometry) and the specific results obtained. The finiteness conjectures on limit cycles are part of a larger picture that also includes finiteness problems in other areas of mathematics, in particular those in Diophantine geometry where remarkable results were proved during the same period of time. There is a chapter devoted to finiteness results in D The volume can be used as an independent study text for advanced undergraduates and graduate students studying ODEs or applications of differential algebra to differential equations and Diophantine geometry. It is also is a good entry point for researchers interested these areas, in particular, in limit cycles of ODEs, and in finiteness problems. Contributors to the volume include Andrey Bolibrukh and Alexandru Buium. Available from the AMS by A. Buium is Arithmetic Differential Equations, as Volume 118 in the Mathematical Surveys and Monographs series.

Sammlung

I. MANIN BIBLIOGRAPHY [ 1 ] I. R. Šafarevič , Principal homogeneous spaces defined over a function field , Trudy Mat . Inst . Steklov . ... [ 10 ] - - , Diophantine equations over functional fields , Dokl . Akad .

Author:

Publisher: World Scientific

ISBN: 9810224982

Category: Mathematics

Page: 616

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The book is a collection of research and review articles in several areas of modern mathematics and mathematical physics published in the span of three decades. The ICM Kyoto talk ?Mathematics as Metaphor? summarises the author's view on mathematics as an outgrowth of natural language.

Number Theory in Function Fields

Madan, M. [1] On class numbers in fields of algebraic functions, Arch. Math, 21 (1970), 161-171. Mason, R.C. [1] The hyperelliptic equation over function fields, Math. Proc. Cambridge Philos. Soc. 93 (1983), 219–230. [2] Diophantine ...

Author: Michael Rosen

Publisher: Springer Science & Business Media

ISBN: 9781475760460

Category: Mathematics

Page: 358

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Early in the development of number theory, it was noticed that the ring of integers has many properties in common with the ring of polynomials over a finite field. The first part of this book illustrates this relationship by presenting analogues of various theorems. The later chapters probe the analogy between global function fields and algebraic number fields. Topics include the ABC-conjecture, Brumer-Stark conjecture, and Drinfeld modules.

New Advances in Transcendence Theory

14 THE STUDY OF DIOPHANTINE EQUATIONS OVER FUNCTION FIELDS R. C. Mason 1. Introduction In recent years there has been important progress in the study of Diophantine equations over function fields . This has aroused increasing interest ...

Author: Alan Baker

Publisher: Cambridge University Press

ISBN: 0521335450

Category: Mathematics

Page: 456

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This is an account of the proceedings of a very successful symposium of Transcendental Number Theory held in Durham in 1986. Most of the leading international specialists were present and the lectures reflected the great advances that have taken place in this area. The papers cover all the main branches of the subject, and include not only definitive research but valuable survey articles.

Value Distribution Theory and Complex Dynamics

[17] Lang, S., Old and new conjectured Diophantine inequalities, Bull. Amer. Math. Soc. 23 (1990), 37–75. MR 90k:11032 [18] Mason, R. C., The hyperelliptic equation over function fields, Math. Proc. Cambridge Philos. Soc.

Author: Kiyoshi L Igusa

Publisher: American Mathematical Soc.

ISBN: 9780821829806

Category: Mathematics

Page: 136

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This volume contains six detailed papers written by participants of the special session on value distribution theory and complex dynamics held in Hong Kong at the First Joint International Meeting of the AMS and the Hong Kong Mathematical Society in December 2000. It demonstrates the strong interconnections between the two fields and introduces recent progress of leading researchers from Asia. In the book, W. Bergweiler discusses proper analytic maps with one critical point and generalizes a previous result concerning Leau domains. W. Cherry and J. Wang discuss non-Archimedean analogs of Picard's theorems. P.-C. Hu and C.-C. Yang give a survey of results in non-Archimedean value distribution theory related to unique range sets, the $abc$-conjecture, and Shiffman's conjecture. L. Keen and J. Kotus explore the dynamics of the family of $f_\lambda(z)=\lambda\tan(z)$ and show that it has much in common with the dynamics of the familiar quadratic family $f_c(z)=z^2+c$. R. Oudkerk discusses the interesting phenomenon known as parabolic implosion and, in particular, shows the persistence of Fatou coordinates under perturbation. Finally, M. Taniguchi discusses deformation spaces of entire functions and their combinatorial structure of singularities of the functions. The book is intended for graduate students and research mathematicians interested in complex dynamics, function theory, and non-Archimedean function theory.

Discriminant Equations in Diophantine Number Theory

Liang, J. (1976), On the integral basis of the maximal real subfield of a cyclotomic field, J. Reine Angew. Math. 286–287, 223–226. ... Mason, R.C. (1984), Diophantine equations over function fields, Cambridge University Press.

Author: Jan-Hendrik Evertse

Publisher: Cambridge University Press

ISBN: 9781316727812

Category: Mathematics

Page:

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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.

Algebra Logic and Combinatorics

Diophantine equations over function fields If k is an algebraically closed field, a function field K over k (of transcendence degree 1) is the field of rational functions of a variety (of dimension 1) over k. An old adage in number ...

Author: Shaun Bullett

Publisher: World Scientific

ISBN: 9781786340320

Category: Mathematics

Page: 184

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This book leads readers from a basic foundation to an advanced level understanding of algebra, logic and combinatorics. Perfect for graduate or PhD mathematical-science students looking for help in understanding the fundamentals of the topic, it also explores more specific areas such as invariant theory of finite groups, model theory, and enumerative combinatorics. Algebra, Logic and Combinatorics is the third volume of the LTCC Advanced Mathematics Series. This series is the first to provide advanced introductions to mathematical science topics to advanced students of mathematics. Edited by the three joint heads of the London Taught Course Centre for PhD Students in the Mathematical Sciences (LTCC), each book supports readers in broadening their mathematical knowledge outside of their immediate research disciplines while also covering specialized key areas. Contents:Enumerative Combinatorics (Peter J Cameron)Introduction to the Finite Simple Groups (Robert A Wilson)Introduction to Representations of Algebras and Quivers (Anton Cox)The Invariant Theory of Finite Groups (Peter Fleischmann and James Shank)Model Theory (Ivan Tomašić) Readership: Researchers, graduate or PhD mathematical-science students who require a reference book that covers algebra, logic or combinatorics.