Uniformly counting points of bounded height

Abstract

1. Introduction. In this paper we give some new uniform estimates for the cardinalities of certain sets involving algebraic numbers of bounded height. The estimates are nearly optimal with respect to the degree of the number eld. We mention some applications to problems about multiplicatively independent and dependent numbers in situations occurring in the recent theory of linear forms in logarithms associated with the name of Matveev. We also extend our counting estimates to algebraic vectors. Let K be a number eld of degree d = [K :Q] over the rational eld Q. We use the absolute height function dened for in K by