On the linear independence of algebraic numbers
نویسندگان
چکیده
منابع مشابه
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15 صفحه اولLinear Independence Measures for Logarithms of Algebraic Numbers
Hence we are back to the problem of estimating from below the distance between 1 and a number of the form α1 1 · · ·αbn n . The first three lectures are devoted to the qualitative theory of transcendental numbers, the last three ones to the quantitative theory of Diophantine approximation. According to Hermite-Lindemann’s Theorem, a number Λ = β − logα, with algebraic α and β, is zero only in t...
متن کاملOn the linear independence measure of logarithms of rational numbers
In this paper we give a general theorem on the linear independence measure of logarithms of rational numbers and, in particular, the linear independence measure of 1, log 2, log 3, log 5 and of 1, log 2, log 3, log 5, log 7. We also give a method to search for polynomials of smallest norm on a real interval [a, b] which may be suitable for computing or improving the linear independence measure ...
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For proving linear independence of real numbers, Hermite [6] considered simultaneous approximation to these numbers by algebraic numbers. The point of view introduced by Siegel in 1929 [14] is dual (duality in the sense of convex bodies): he considers simultaneous approximation by means of independent linear forms. We define the height of a linear form L = a0X0 + · · · + amXm with complex coeff...
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Let n be a positive integer. Let ξ be an algebraic real number of degree greater than n. It follows from a deep result of W. M. Schmidt that, for every positive real number ε, there are infinitely many algebraic numbers α of degree at most n such that |ξ−α| < H(α)−n−1+ε, where H(α) denotes the näıve height of α. We sharpen this result by replacing ε by a function H 7→ ε(H) that tends to zero wh...
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ژورنال
عنوان ژورنال: Pacific Journal of Mathematics
سال: 1953
ISSN: 0030-8730,0030-8730
DOI: 10.2140/pjm.1953.3.625