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Effective Results and Methods for Diophantine Equations over Finitely Generated Domains
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Artikel-Nr:
9781009005852
Veröffentl:
2022
Erscheinungsdatum:
28.04.2022
Seiten:
240
Autor:
Jan-Hendrik Evertse
Gewicht:
376 g
Format:
229x153x18 mm
Sprache:
Englisch
Beschreibung:
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 Gy¿ry: Unit Equations in Diophantine Number Theory (Cambridge, 2016) and Discriminant Equations in Diophantine Number Theory (Cambridge, 2016).
"This book is devoted to Diophantine equations where the solutions are taken from an integral domain of characteristic 0 that is finitely generated over Z, that is a domain of the shape Z[z1; :: zr] with quotient field of characteristic 0, where the generators z1; :: zr may be algebraic or transcendental over Q. For instance, the ring of integers and the rings of S-integers of a number field are finitely generated domains where all generators are algebraic. Our aim is to prove effective finiteness results for certain classes of Diophantine equations, i.e., results that not only show that the equations from the said classes have only finitely many solutions, but whose proofs provide methods to determine the solutions in principle"--
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.