## 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:

  Bundschuh, P.: Ma?e f?r die lineare Unabh?ngigkeit gewisser Zahlen. Tagungsbericht Math. Forsch. Inst. Oberwolfach30, 3-4 (1974).  Bundschuh, P.: Einf?hrung in die Zahlentheorie. (2. Aufl.) Berlin: Springer. 1992. · Zbl 0746.11001  Fel’dman, N. I., Shidlovskii, A. B.: The development and present state of the theory of transcendental numbers (Russian). Uspekhi Mat. Nauk22, 3-81 (1967). Engl. transl.: Russian Math. Surveys22, 1-79 (1967).  Mahler, K.: Zur Approximation der Exponentialfunktion und des Logarithmus. I. J. Reine Angew. Math.166, 118-136 (1932). · JFM 57.0242.03  Nesterenko, Y. V.: On the linear independence of numbers (Russian). Vestnik Moskov. Univ. Ser. I Mat. Mekh.1, 46-49 (1985). Engl. transl.: Moscow Univ. Math. Bull.40, 69-74 (1985).  Philippon, P.: Crit?res pour l’ind?pendance alg?brique. Inst. Hautes Etudes Sci. Publ. Math.64, 5-52 (1986). · Zbl 0615.10044  Popken, J.: Zur Transzendenz vone. Math. Z.29, 525-541 (1929). · JFM 55.0117.01  Shidlovskii, A. B.: Transcendental Numbers. Berlin: De Gruyter. 1989. · Zbl 0689.10043
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