Effective Results and Methods for Diophantine Equations over Finitely Generated Domains

Erscheinungsjahr 2022
ISBN: 978-1-009-00585-2
Verlag: Cambridge University Press

Buch, Englisch, Band 475, 300 Seiten, Format (B × H): 229 mm x 153 mm, Gewicht: 376 g

Reihe: London Mathematical Society Lecture Note Series

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.

Autoren/Hrsg.

Evertse, Jan-Hendrik

Gyory, Kalman

Gyory, Kálmán

Fachgebiete

• Mathematik | Informatik Mathematik Algebra Zahlentheorie
• Mathematik | Informatik Mathematik Geometrie Algebraische Geometrie

Inhaltsverzeichnis

Preface; Glossary of frequently used notation; History and summary; 1. Ineffective results for Diophantine equations over finitely generated domains; 2. Effective results for Diophantine equations over finitely generated domains: the statements; 3. A brief explanation of our effective methods over finitely generated domains; 4. Effective results over number fields; 5. Effective results over function fields; 6. Tools from effective commutative algebra; 7. The effective specialization method; 8. Degree-height estimates; 9. Proofs of the results from Sections 2.2–2.5-use of specializations; 10. Proofs of the results from Sections 2.6–2.8-reduction to unit equations; References; Index.

Über Autor(innen)

Gyory, Kálmán
Kálmán Gyory is Professor Emeritus at the University of Debrecen, Hungary and a member of the Hungarian Academy of Sciences. Gyory is the founder and leader of the Number Theory Research Group in Debrecen. Together with Jan-Hendrik Evertse he has written two books: Unit Equations in Diophantine Number Theory (Cambridge, 2016) and Discriminant Equations in Diophantine Number Theory (Cambridge, 2016).

Evertse, Jan-Hendrik
Jan-Hendrik Evertse is Associate Professor in Number Theory at Leiden University in the Netherlands. He co-edited the lecture notes in mathematics Diophantine Approximation and Abelian Varieties (1993) with Bas Edixhoven, and co-authored two books with Kálmán Gyory: Unit Equations in Diophantine Number Theory (Cambridge, 2016) and Discriminant Equations in Diophantine Number Theory (Cambridge, 2016).