Some cases of Wilkie's conjecture
type: Item
instance of: scholarly article
title: Some cases of Wilkie's conjecture
zbMATH Open document ID: 1253.03063
author: Lee A. Butler
published in: Bulletin of the London Mathematical Society
publication date: 2012-08-08T00:00:00Z
Mathematics Subject Classification ID: 03C64
Mathematics Subject Classification ID: 11G99
zbMATH DE Number: 6064189
zbMATH Keywords: definable set
zbMATH Keywords: rational point
zbMATH Keywords: Wilkie's conjecture
Wikidata QID: Q123018219
MaRDI profile type: Publication
full work available at URL: bdr126
OpenAlex ID: W1977225932
DOI: 10.1112/BLMS/BDR126
review text: Wilkie has conjectured a very strong upper bound for the number of the rational points up to a given height on a set definable in the o-minimal structure \({\mathbb R}_{\exp}\) that do not lie on some positive-dimensional semialgebraic subset. Specifically, a bound polynomial in the logarithm of the height. This paper affirms the conjecture for curves (i.e., definable sets of dimension 1) in this structure, and for certain surfaces. For the case of surfaces, the main issue is to establish that they can be parameterised in a suitably controlled way. The case of curves was obtained independently by \textit{G. O. Jones} and \textit{M. E. M. Thomas} [Q. J. Math. 63, No. 3, 637--651 (2012; Zbl 1253.03065)], who also affirm the conjecture for some different surfaces.
reviewed by: Jonathan Pila
description: scientific article; zbMATH DE number 6064189