Counting rational points on a certain exponential-algebraic surface
type: Item
instance of: scholarly article
title: Counting rational points on a certain exponential-algebraic surface
zbMATH Open document ID: 1210.11074
author: Jonathan Pila
publication date: 2010-06-25T00:00:00Z
full work available at URL: https://eudml.org/doc/116279
review text: Let \({\mathcal X}=\{(x,y,z)\in (0,1)^3:\log x \log y=-\log z\}.\) Let \(F\subset{\mathbb R}\) be a number field of degree \(f\) over \(\mathbb Q\) and let \(\varepsilon>0\). The author proves that there exists a constant \(c({\mathcal X},f,\varepsilon)>0 \) such that \[ \#{\mathcal X}(F,T)\leq c({\mathcal X},f,\varepsilon)(\log T)^{44+\varepsilon}, \] where \({\mathcal X}(F,T)\) denotes the set of \(F\)-rational points of \({\mathcal X}\) with height \(\leq T \).
reviewed by: Florin Nicolae
Mathematics Subject Classification ID: 11G50
Mathematics Subject Classification ID: 03C64
Mathematics Subject Classification ID: 14G05
Mathematics Subject Classification ID: 11J91
zbMATH DE Number: 5726202
zbMATH Keywords: rational points
zbMATH Keywords: exponential-algebraic surface
MaRDI profile type: Publication
OpenAlex ID: W2318780325
cites work: Algebraic values of meromorphic maps
cites work: Q5719099
cites work: The number of integral points on arcs and ovals
cites work: Some cases of Wilkie's conjecture
cites work: Q4392286
cites work: Geometric categories and o-minimal structures
cites work: BETTI NUMBERS OF SEMIALGEBRAIC AND SUB-PFAFFIAN SETS
cites work: Algebraic values of meromorphic functions
cites work: Q5520004
cites work: Integer points on the dilation of a subanalytic surface
cites work: Rational points on a subanalytic surface.
cites work: Q3420204
cites work: The density of rational points on a Pfaff curve
cites work: On the algebraic points of a definable set
cites work: The rational points of a definable set
cites work: Interpolation formulas and auxiliary functions
cites work: Q4353891
cites work: Q3887541
cites work: Q4488162
cites work: Algebraic values of analytic functions.
cites work: Model completeness results for expansions of the ordered field of real numbers by restricted Pfaffian functions and the exponential function
cites work: A theorem of the complement and some new o-minimal structures
DOI: 10.5802/AIF.2530
published in: Annales de l’institut Fourier
description: scientific article; zbMATH DE number 5726202