Algebraic Independence by Approximation Method (II)
Copyright policy )Algebraic Independence by Approximation Method (II)
Summary: Let \(f(x)\) be a continued fraction with elements \(a_nx\), where the coefficients \(a_n\) are positive algebraic numbers. Using the criterion of Part I [Acta Math. Sin., New Ser. 14, 295-302 (1998; Zbl 0920.11047)] for any nonzero real algebraic numbers \(\alpha_1,\dots,\alpha_s\) with distinct absolute values the algebraic independence of the values \(f(\alpha_1),\dots, f(\alpha_s)\) is proved under a certain assumption concerning only \(a_n\). For some transcendental numbers \(\xi\) the algebraic independence of the values \(f(\xi^j)\) \((j\in\mathbb{Z})\) is also established. The English version appeared in Acta Math. Sin., Eng. Ser. 16, 395-398 (2000; Zbl 0961.11025).
- Academia Sinica Taiwan
- Chinese Academy of Sciences China (People's Republic of)
- Institute of Applied Physics and Computational Mathematics China (People's Republic of)
- Academy of Mathematics and Systems Science China (People's Republic of)
Algebraic independence; Gel'fond's method, algebraic independence, Continued fractions, approximation method, transcendence measure, Transcendence theory of other special functions, continued fraction, transcendental numbers
Algebraic independence; Gel'fond's method, algebraic independence, Continued fractions, approximation method, transcendence measure, Transcendence theory of other special functions, continued fraction, transcendental numbers
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