## 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

Zbl 0572.10027
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### References:

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