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Counting Algebraic Numbers with Large Height I

  • David Masser
  • Jeffrey D. Vaaler
Part of the Developments in Mathematics book series (DEVM, volume 16)

Abstract

Let ℚ denote the field of rational numbers, Open image in new window an algebraic closure of ℚ, and H : Open image in new window the absolute, multiplicative, Weil height. For each positive integer d and real number \( \mathcal{H} \geqslant 1 \), it is well known that the number Open image in new window of points α in Open image in new window having degree d over ℚ and satisfying \( H\left( \alpha \right) \leqslant \mathcal{H} \) is finite. This is the one-dimensional case of Northcott’s Theorem [8] (see also [5, page 59]). The systematic study of the counting function Open image in new window , and that of related functions in higher dimensions, was begun by Schmidt [10]. It is relatively easy to prove the existence of a positive constant C = C(d) such that and also the existence of positive constants c = c(d) and \( \mathcal{H}_0 = \mathcal{H}_0 \left( d \right) \) such that

Keywords

Mahler measure height 

2000 Mathematics subject classification

11R04 

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Copyright information

© Springer-Verlag 2008

Authors and Affiliations

  • David Masser
    • 1
  • Jeffrey D. Vaaler
    • 2
  1. 1.Mathematisches InstitutUniversität BaselBaselSwitzerland
  2. 2.Department of MathematicsUniversity of Texas at AustinAustinUSA

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