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On linear independence. (Über lineare Unabhängigkeit.) (German) Zbl 0804.11044

An efficient method of elimination has been introduced by Yu. V. Nesterenko and yields criteria of algebraic independence; it is useful for proving a lower bound for the transcendence degree over the rational number field \(\mathbb{Q}\) of fields generated by values of certain functions. In the linear case, when only polynomials of degree 1 are concerned, the method can be pushed further and yields sharp criteria of linear independence; it provides lower bounds for the dimension of \(\mathbb{Q}\)- vector subspaces generated by values of analytic functions [Yu. V. Nesterenko, Mosc. Univ. Math. Bull. 40, 69-74 (1985); translation from Vestn. Mosk. Univ. Ser. I, 46-49 (1985; Zbl 0572.10027)]. The authors improve Nesterenko’s criterion for linear independence and give a quantitative version of it, which is useful for proving sharp measures of linear independence. They provide an application concerning the entire transcendental functions.
\[ \varphi_ \lambda (z) = \sum_{n \geq 0} z^ n \prod_{\nu = 1}^ n (\nu + \lambda)^{-1}, \] for \(\lambda \in \mathbb{Q}\) with \(\lambda \notin \{-1,-2, \dots\}\). Let \(r_ 1, \dots, r_ s\) be pairwise distinct nonzero rational numbers. Then the numbers \(\varphi_ \lambda (r_ 1), \dots, \varphi_ \lambda (r_ s)\) are linearly independent over \(\mathbb{Q}\) and satisfy the following measure of linear independence: there is a constant \(\gamma\), which depends only on \(\lambda, r_ 1, \dots, r_ s\), such that for any linear form \(L\) in \(s+1\) variables with Euclidean norm \(\| L \| \geq 3\), \[ \biggl | L \bigl( 1, \varphi_ \lambda (r_ 1), \dots, \varphi_ \lambda (r_ s) \bigr) \biggr | \geq \| L \|^{-s - \gamma/ \log \log \| L \|}. \]

MSC:

11J82 Measures of irrationality and of transcendence
11J72 Irrationality; linear independence over a field

Citations:

Zbl 0572.10027
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References:

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