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Heights of Algebraic Numbers

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Part of the Grundlehren der mathematischen Wissenschaften book series (GL, volume 326)

Abstract

A nonzero rational integer has absolute value at least 1. A nonzero rational number has absolute value at least the inverse of any denominator. Liouville’s inequality (§ 3.5) is an extension of these estimates and provides a lower bound for the absolute value of any nonzero algebraic number. More specifically, if we are given finitely many (fixed) algebraic numbers γ l,...,γ t , and a polynomial P ∈ ℤ[X1,...,X t ] which does not vanish at the point (γ l,...,γ t ) then we can estimate from below |P(γ l,...,γ t )|. The lower bound will depend upon the degrees of P with respect to each of the X i ’s, the absolute values of its coefficients as well as some measure of the γ i ’s.

Keywords

Number Field Algebraic Number Minimal Polynomial Product Formula Algebraic Integer 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2000

Authors and Affiliations

  1. 1.Institut de Mathématiques de JussieuUniversité Pierre et Marie Curie (Paris VI)Paris Cedex 05France

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