Lecture notes on Diophantine analysis. With an appendix by Francesco Amoroso (Q6594824)

This is the second edition of the author's book (for the first edition see [Lecture notes on Diophantine analysis. With an appendix by Francesco Amoroso. Pisa: Edizioni della Normale (2009; Zbl 1186.11001)] on a set of lecture notes on diophantine analysis. In addition to classical diophantine results, the author includes the basic theory of heights and a few more recent results and their applications.\N\NThe book is intended to all interested number theorists, not necessarily in this area, as well as specialists on diophantine equations. Therefore most of the text is on elementary self-contained level, giving an interesting introduction to the topic. In addition, several recent results and methods are also involved, for the interest of specialists.\N\NThis second edition provides several new material, as new sections, supplements, remarks, and exercises.\N\NIn Chapter 1 linear equations, quadratic equations, the Pell equation are considered, and their relation to diophantine approximation is described.\N\NChapter 2 discusses the Thue equation, with applications of diophantine approximation, Thue's theorem with sharpenings, like Roth theorem. All equations are considered over \(\mathbb Z\) to make it understandable for beginners.\N\NChapter 3 gives diophantine problems over number fields, discusses the Weil height and its properties, mentions the general Roth theorem with application to the Thue-Mahler equation and S-unit equation. Heights on finitely generated multiplicative groups of algebraic numbers are investigated.\N\NIn Chapter 4 the author considers diophantine problems with variables in \(\overline{\mathbb Q}\), subject to arithmetical restrictions (e.g. solutions in roots of unity of arbitrary order). An elementary proof of Zhang's theorem is given with an independent elementary approach by Bilu.\N\NIn Chapter 5 we return to S-unit equations with several applications, with a complete independent proof of the S-unit theorem by Beukers and Schlickewei (giving estimate for the number of solutions). This is applied to Thue-Mahler equations.\N\NChapters 1--4 are finished with supplements containing related material among others on the irrationality of \(e^{\pi}\) and \(\pi\) (Ch. 1), the geometry of algebraic curves (Ch. 2), S-unit equations over function fields (Ch. 3), a basic theory of closed subgroups of \(\mathbb R^n\) and the Skolem-Mahler-Lech theorem on zeros of recurrences (Ch. 4).\N\NThe book closed with the Appendix by F. Amoroso, listing problems of Chapter 4 from a quantitative point of view with several new results and sketches of methods of proofs.\N\NIt turns out again, that Thue's equation is the best example to discuss most of the diophantine techniques. The book will certainly wake the interest of all number theorists.