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

Bundschuh, Peter; Töpfer, Thomas

Monatsh. Math. 117, No. 1-2, 17-32 (1994).

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 \(\| L \| \geq 3\), \[ \biggl | L \bigl( 1, \varphi_ \lambda (r_ 1), \dots, \varphi_ \lambda (r_ s) \bigr) \biggr | \geq \| L \|^{-s - \gamma/ \log \log \| L \|}. \]

Reviewer: M.Waldschmidt (Paris)

MSC:

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

Keywords:

Nesterenko’s criterion for linear independence; quantitative version; sharp measures of linear independence; entire transcendental functions

Citations:

Zbl 0572.10027

References:

[1]	Bundschuh, P.: Ma?e f?r die lineare Unabh?ngigkeit gewisser Zahlen. Tagungsbericht Math. Forsch. Inst. Oberwolfach30, 3-4 (1974).
[2]	Bundschuh, P.: Einf?hrung in die Zahlentheorie. (2. Aufl.) Berlin: Springer. 1992. · Zbl 0746.11001
[3]	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).
[4]	Mahler, K.: Zur Approximation der Exponentialfunktion und des Logarithmus. I. J. Reine Angew. Math.166, 118-136 (1932). · JFM 57.0242.03
[5]	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).
[6]	Philippon, P.: Crit?res pour l’ind?pendance alg?brique. Inst. Hautes Etudes Sci. Publ. Math.64, 5-52 (1986). · Zbl 0615.10044
[7]	Popken, J.: Zur Transzendenz vone. Math. Z.29, 525-541 (1929). · JFM 55.0117.01
[8]	Shidlovskii, A. B.: Transcendental Numbers. Berlin: De Gruyter. 1989. · Zbl 0689.10043