Introduction to Algebraic Independence Theory

Volume 1752 of the series Lecture Notes in Mathematics pp 227-237


Some metric results in Transcendental Numbers Theory

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In this Chapter we describe some results in the metric theory of transcendental numbers. Let begin with some notation. If P ∈ Z[x[inl, ⋯, x m] is a non - zero polynomial, we define its size t(P) as h(P) + deg (P). Here, h(P) is the Weil’s logarithmic height of P (so, if the ged of the coefficients of P is 1, then h(P) is the logarithm of the maximum module of the coefficients of P) and deg (P) is the total degree of P. Let α = (α1,⋯, αm) ∈ Cm with α1,⋯, αm algebraically dependent: we define t(α) as the minimum size of a non - zero polynomial P ∈ Z[x l⋯,x m] such that P(α) = 0. er’s author : Francesco AMOROSO.