Mildness and the density of rational points on certain transcendental curves
type: Item
instance of: scholarly article
title: Mildness and the density of rational points on certain transcendental curves
zbMATH Open document ID: 1220.03034
DOI: 10.1215/00294527-2010-037
author: G. O. Jones
author: Daniel J. Miller
author: M. E. M. Thomas
published in: Notre Dame Journal of Formal Logic
publication date: 2011-03-07T00:00:00Z
review text: This paper concerns the density of rational points on certain real-analytic non-algebraic plane curves. The curves in question are restricted Pfaffian, namely, one-dimensional subsets of the plane definable in the o-minimal structure \({\mathbf R}_{\text{Res\,Pfaff}}\). The main theorem asserts that, for a non-algebraic curve \(X\) of this type, the number \(N(X,T)\) of rational points on the curve having both coordinates of height bounded by \(H\) is bounded by \(c(\log H)^\gamma\) for suitable constants \(c,\gamma\) depending on \(X\). The proof depends on the possibility of parameterising the curve in a suitable way, and is obtained by modifying the approach of \textit{J. Pila} [Comment. Math. Univ. St. Pauli 55, No.~1, 1--8 (2006; Zbl 1129.11029)], where similar results are obtained for a more restricted type of curve definable in \({\mathbf R}_{\text{Pfaff}}\).
reviewed by: Jonathan Pila
Mathematics Subject Classification ID: 03C64
Mathematics Subject Classification ID: 11G99
Mathematics Subject Classification ID: 11U09
zbMATH DE Number: 5862016
zbMATH Keywords: Pfaffian functions
zbMATH Keywords: parameterization
zbMATH Keywords: rational point
zbMATH Keywords: o-minimal structure
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1215/00294527-2010-037
OpenAlex ID: W2117551059
description: scientific article; zbMATH DE number 5862016