Explicit uniform estimation of rational points. II: Hypersurface coverings
type: Item
instance of: scholarly article
zbMATH Open document ID: 1248.14031
DOI: 10.1515/CRELLE.2011.139
author: Huayi Chen
publication date: 2012-08-04T00:00:00Z
Mathematics Subject Classification ID: 14G40
Mathematics Subject Classification ID: 11D45
Mathematics Subject Classification ID: 14G05
Mathematics Subject Classification ID: 11G50
zbMATH DE Number: 6063236
zbMATH Keywords: Diophantine geometry
zbMATH Keywords: counting rational points
zbMATH Keywords: number fields
zbMATH Keywords: bounded height
zbMATH Keywords: Arakelov geometry
zbMATH Keywords: slope method
zbMATH Keywords: determinant method
MaRDI profile type: Publication
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title: Explicit uniform estimation of rational points. II: Hypersurface coverings
published in: Journal für die Reine und Angewandte Mathematik
review text: The purpose of these two papers (for part I, cf. [ibid. 668, 59--88 (2012; Zbl 1249.14010)]) is to develop the ``determinant method'' due to \textit{E. Bombieri} and \textit{J. Pila} [Duke Math. J. 59, No. 2, 337--357 (1989; Zbl 0718.11048)] and extended by the reviewer [Ann. Math. (2) 155, No. 2, 553--598 (2002; Zbl 1039.11044)]. The primary new idea is to interpret the method within the context of Arakelov geometry, and to use \textit{J.-B. Bost}'s slope theory, see for example [Proceedings of the international congress of mathematicians (ICM), Madrid, Spain, August 22--30, 2006. Volume II: Invited lectures. Zürich: European Mathematical Society (EMS). 537--562 (2006; Zbl 1141.14011)]. For a number field \(K\) the outcome is a result for a general integral closed subvariety \(X\) of \(\mathbb{P}_K^n\) of dimension \(d\) and degree \(\delta\). Given any \(\varepsilon>0\) take \(D\) to be the smallest integer greater than NEWLINE\[NEWLINE\max\{(\varepsilon^{-1}+1)(2\delta^{-1/d}(d+1)+\delta-2)\,,\, 2(n-d)(\delta-1)+d+2\}.NEWLINE\]NEWLINE Then there is an explicitly computable constant \(C=C(\varepsilon,\delta,n,d,K)\) such that if \(B\geq e^{\varepsilon}\) the rational points of \(X\) of exponential height at most \(B\) are covered by at most NEWLINE\[NEWLINECB^{(1+\varepsilon)\delta^{-1/d}(d+1)}NEWLINE\]NEWLINE hypersurfaces of degree \(\leq D\) not containing \(X\). In particular the degrees which occur are \(O_n(\varepsilon^{-1}\delta)\).NEWLINENEWLINEThe interest of this result is firstly that it applies to general varieties \(X\) rather than merely to hypersurfaces, and secondly that the bound \(D\) on the degrees is completely explicit. \textit{N. Broberg} [J. Reine Angew. Math. 571, 159--178 (2004; Zbl 1053.11027)], had previously given a result for general varieties which depended on the degrees of the forms in the defining ideal. The present paper removes this dependence. In the case of an affine plane curve over the rationals, \textit{Y. Walkowiak} [Acta A…
reviewed by: D. R. Heath-Brown
description: scientific article; zbMATH DE number 6063236