Diophantine approximation with algebraic points of bounded degree.
type: Item
instance of: scholarly article
title: Diophantine approximation with algebraic points of bounded degree.
zbMATH Open document ID: 1049.11070
author: Min Ru
author: Julie Tzu-Yueh Wang
published in: Journal of Number Theory
publication date: 2000-06-07T00:00:00Z
review text: The authors extend Schmidt's subspace theorem to the approximation of algebraic numbers by algebraic points of bounded degree. Earlier E. Wirsing proved an extension of Roth's theorem that was generalized by W. M. Schmidt to the approximation of algebraic hyperplanes in projective space by \(k\)-points, \(k\) is a fixed number field. Now here the authors prove as their main theorem a Wirsing-style generalization of Schmidt's theorem. Their method of proof is similar to the so-called ``Stoll's algebroid reduction method'' [cf. \textit{W. Stoll}, Lect. Notes Math. 1277, 131--241 (1987; Zbl 0627.32002)] in Nevanlinna theory and exposed in another paper of \textit{Min Ru} [Math. Z. 233, 137--148 (2000; Zbl 0951.32015)]. Finally they apply their main theorem to solutions of an equation of the form \(F(\mathbf x) = G(\mathbf x),\) where \(F\) and \(G\) are homogeneous forms with \(S\)-integral coefficients in \(n+1\) variables, where \(S\) is a finite set of places containing all the Archimedean places in a number field \(k\). If \(F\) decomposes into a product of linear forms satisfying the strong general position hypothesis of their main theorem, and if \(\deg G < \deg F - 2 {{n+r}\choose r} + 2,\) then there are only finitely many algebraic \(S\)-integer solutions \(\mathbf x\) to \(F(\mathbf x) = G( \mathbf x)\) with \([k(\mathbf x): k]\leq r\).
reviewed by: Olaf Ninnemann
Mathematics Subject Classification ID: 11J13
Mathematics Subject Classification ID: 11J25
Mathematics Subject Classification ID: 11J97
Mathematics Subject Classification ID: 32H30
zbMATH DE Number: 1420266
MaRDI profile type: Publication
full work available at URL: jnth.1999.2462
OpenAlex ID: W2081974039
cites work: Diophantine approximation on abelian varieties
cites work: Q3309989
cites work: Algebroid functions, Wirsing's theorem and their relations
cites work: Integral points of \(\mathbb{P}^ n-\{2n+1\) hyperplanes in general position\(\}\)
cites work: Diophantine approximation
cites work: Q3762831
cites work: Diophantine approximations and value distribution theory
cites work: A Generalization of Theorems of Faltings and Thue-Siegel-Roth-Wirsing
cites work: Q4237133
cites work: Q5628199
DOI: 10.1006/JNTH.1999.2462
Recommended article: An effective Schmidt’s subspace theorem for arbitrary hypersurfaces over function fields
Recommended article: A generalization of the subspace theorem for higher degree polynomials in subgeneral position
Recommended article: A Wirsing-type theorem for numerically equivalent divisors
Recommended article: Wirsing-type inequalities
Recommended article: Q4673155
Recommended article: Subspace theorems for homogeneous polynomial forms
Recommended article: Q4332779
Recommended article: A further improvement of the quantitative subspace theorem
Recommended article: An effective function field version of Schmidt's subspace theorem for projective varieties, with arbitrary families of homogeneous polynomials
Recommended article: Q4732523
description: scientific article; zbMATH DE number 1420266