Lecture Notes on Diophantine Analysis
Published on 31. March 2009
Book
Paperback/Softback
XVI, 239 pages
978-88-7642-341-3 (ISBN)
Description
These lecture notes originate from a course delivered at the Scuola Normale in Pisa in 2006. Generally speaking, the prerequisites do not go beyond basic mathematical material and are accessible to many undergraduates. The contents mainly concern diophantine problems on affine curves, in practice describing the integer solutions of equations in two variables. This case historically suggested some major ideas for more general problems. Starting with linear and quadratic equations, the important connections with Diophantine Approximation are presented and Thue's celebrated results are proved in full detail. In later chapters more modern issues on heights of algebraic points are dealt with, and applied to a sharp quantitative treatment of the unit equation. The book also contains several supplements, hinted exercises and an appendix on recent work on heights.
Reviews / Votes
From the reviews:
"This set of lectures originates from an introductory course given by the author at the Scuola Normale Superiore Pisa in 2006-2007 on Diophantine analysis. . provide a lot of further information on these topics, including recent developments and open problem. Hence this booklet starts at an elementary level . and reaches the limit of our current knowledge on all these topics. . a useful addition to the literature on this fashionable topic, both for the beginner and for the researcher." (Michel Waldschmidt, Zentralblatt MATH, Vol. 1186, 2010)More details
Series
Edition
1st ed. 2009. Corr. 3rd printing 2014
Language
English
Place of publication
Pisa
Switzerland
Publishing group
Birkhauser Verlag AG
Target group
Primary & secondary/elementary & high school
Graduate
Illustrations
XVI, 239 p.
Dimensions
Height: 240 mm
Width: 150 mm
Thickness: 15 mm
Weight
389 gr
ISBN-13
978-88-7642-341-3 (9788876423413)
DOI
10.1007/978-88-7642-517-2
Schweitzer Classification
Persons
Content
1. Classical Diophantine Equations: linear and quadratic equations, Pell Equation, Diophantine Approximation, congruences. Supplements on Pell equations and irrationality of exp(n) and pi. Notes.- 2. Thue's theorems on Diophantine Equations and rational approximations: Description of strategy and detailed proofs. Later refinements. Supplements on integral points on curves and Runge's theorem. Notes.- 3. Heights and Diophantine equations over number fields: Product formulas, Weil and Mahler heights, Diophantine approximation in number fields, the S-unit equation and its applications. Supplements on the abc-theorem in function fields and on multiplicative dependence of algebraic functions and their values. Notes.- 4. Heights on subvarieties of G_m^n: Torsion points on plane curves and algebraic points of small height on subvarieties of G_m^n. Structure of algebraic subgroups. Theorems of Zhang and Bilu and applications to the S-unit equation. Supplements on discrete and closed subgroups of R^n and on the Skolem-Mahler-Lech theorem. Notes.- 5. The S-unit equation. A sharp quantitative S-unit theorem; explicit Pade' approximations and the counting of large solutions; counting of small solutions. Applications of the quantitative S-unit theorem. Notes.- Appendix by F. Amoroso: Bounds for the height: Generalized Lehmer problem, Dobrowolski lower bounds. Heights of varieties and extensions of lower bounds to higher dimensions; sharp quantitative Zhang's theorem.