Diophantine Equations and Power Integral Bases: Theory and Algorithms

ISBN-13: 9783030238643 / Angielski / Twarda / 2019 / 326 str.

Istvan Gaal

Opis

Work examines the latest algorithms and tools to solve classical types of diophantine equations.; Unique book---closest competitor, Smart, Cambridge, does not treat index form equations.; Author is a leading researcher in the field of computational algebraic number theory.; The text is illustrated with several tables of various number fields, including their data on power integral bases.; Several interesting properties of number fields are examined.; Some infinite parametric families of fields are also considered as well as the resolution of the corresponding infinite parametric families of diophantine equations.

Recenzje

"The book is recommended to PhD students and researchers in the field of Diophantine equations. It can be used as a textbook for a specialized graduate course in Thue and index-form equations and as an additional reading for a general course in Diophantine equations." (Andrej Dujella, zbMATH 1465.11090, 2021)

Spis treści

Introduction.- Auxiliary results, tools.- Thue equations.- Inhomogeneous Thue equations.- Relative Thue equations.- The resolution of norm form equations.- Index form equations in general.- Cubic fields.- Quartic fields.- Quintic fields.- Sextic fields.- Pure fields.- Cubic relative extensions.- Quartic relative extensions.- Some higher degree fields.- Tables.

Blurb

This monograph outlines the structure of index form equations, and makes clear their relationship to other classical types of Diophantine equations. In order to more efficiently determine generators of power integral bases, several algorithms and methods are presented to readers, many of which are new developments in the field. Additionally, readers are presented with various types of number fields to better facilitate their understanding of how index form equations can be solved. By introducing methods like Baker-type estimates, reduction methods, and enumeration algorithms, the material can be applied to a wide variety of Diophantine equations. This new edition provides new results, more topics, and an expanded perspective on algebraic number theory and Diophantine Analysis.

Notations, definitions, and tools are presented before moving on to applications to Thue equations and norm form equations. The structure of index forms is explained, which allows readers to approach several types of number fields with ease. Detailed numerical examples, particularly the tables of data calculated by the presented methods at the end of the book, will help readers see how the material can be applied.

Diophantine Equations and Power Integral Bases will be ideal for graduate students and researchers interested in the area. A basic understanding of number fields and algebraic methods to solve Diophantine equations is required.