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001-es BibID:	BIBFORM103304
Első szerző:	Evertse, Jan-Hendrik
Cím:	Mahler's Work on Diophantine Equations and Subsequent Developments / Evertse, Jan-Hendrik, Győry Kálmán, Stewart, Cameron L.
Dátum:	2019
Megjegyzések:	The main body of Mahler's work on Diophantine equations consists of his 1933 papers [M17, M18, M19], in which he proved a generalisation of the Thue-Siegel Theorem on the approximation of algebraic numbers by rationals, involving P-adic absolute values, and applied this to get finiteness results for the number of solutions for what became later known as Thue-Mahler equations. He was also the first to give upper bounds for the number of solutions of such equations. In fact, Mahler's extension of the Thue-Siegel Theorem made it possible to extend various finiteness results for Diophantine equations over the integers to S-integers, for any arbitrary finite set of primes S. For instance Mahler himself [M21] extended Siegel's finiteness theorem on integral points on elliptic curves to S-integral points. In this chapter, we discuss Mahler's work on Diophantine approximation and its applications to Diophantine equations, in particular Thue-Mahler equations, S-unit equations and S-integral points on elliptic curves, and go into later developments concerning the number of solutions to Thue-Mahler equations and effective finiteness results for Thue-Mahler equations. For the latter we need estimates for P-adic logarithmic forms, which may be viewed as an outgrowth of Mahler's work on the P-adic Gel'fond-Schneider theorem [M30]. We also go briefly into decomposable form equations, these are certain higher dimensional generalisations of Thue-Mahler equations.
Tárgyszavak:	Természettudományok Matematika- és számítástudományok idegen nyelvű folyóiratközlemény külföldi lapban folyóiratcikk
Megjelenés:	Documenta Mathematica. - Ext.Vol. Mahl.Sel. (2019), p. 149-171. -
További szerzők:	Győry Kálmán (1940-) (matematikus) Stewart, Cameron Leigh (1949-) (matematikus)
Internet cím:	DOI Intézményi repozitóriumban (DEA) tárolt változat
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