Abstract
It is the aim of the present work to prove, under appropriate conditions, lower estimates for the dimension of ℚw 1 + ... + ℚw m over ℚ, wherew 1,...,w m are given real numbers. In particular, if this dimension ism, i.e. ifw 1,...,w m are linearly independent over ℚ, we are also interested in a quantitative version of this fact. Our qualitative theorems generalize a result of Nesterenko. Its formulation is quite similar to the “axiomatization” of methods for algebraic independence, as it became usual during the last decade.
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Literaturverzeichnis
Bundschuh, P.: Maße für die lineare Unabhängigkeit gewisser Zahlen. Tagungsbericht Math. Forsch. Inst. Oberwolfach30, 3–4 (1974).
Bundschuh, P.: Einführung in die Zahlentheorie. (2. Aufl.) Berlin: Springer. 1992.
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).
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.: Critères pour l'indépendance algébrique. Inst. Hautes Etudes Sci. Publ. Math.64, 5–52 (1986).
Popken, J.: Zur Transzendenz vone. Math. Z.29, 525–541 (1929).
Shidlovskii, A. B.: Transcendental Numbers. Berlin: De Gruyter. 1989.