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A theorem about algebraic independence and its application. (Chinese. English summary) Zbl 0531.10036

Put \(f_{\nu}(z)=\sum^{\infty}_{k=1}\alpha_{\nu,k} z^{\lambda_{\nu,k}}\), \(\nu =1,...,m\), where all the \(\alpha_{\nu,k}\in {\mathbb{Q}}\backslash \{0\}\) and for each \(\nu\), \(\lambda_{\nu,k} (k=1,2,...)\) is a monotonously increasing sequence of natural numbers tending to infinity. Let \(g_{\nu}>1 (\nu =1,...,m)\) be natural numbers and \(\sigma_{\nu,n}=\sum^{n}_{k=1}\alpha_{\nu,k}\quad g_{\nu}^{- \lambda_{\nu,k}},\quad \tau_{\nu,n}=f_{\nu}(g_{\nu}^{-1})- \sigma_{\nu,n}.\) In this article the following is proved. Theorem 1. Suppose that \((i)\quad \lambda_{\nu,n}=o(\lambda_{\mu,n}),\) \(n\to \infty\), \(1\leq \nu<\mu \leq m\), and \(\lambda_{m,n}=o(\lambda_{1,n+1}),\) \(n\to \infty\); (ii) the least common denominator \(d_{\nu,n}\) of \(\alpha_{\nu,1},...,\alpha_{\nu,n}\) satisfies \(\log d_{\nu,n}=o(\lambda_{\nu,n+1}),\) \(n\to \infty\), \(\nu =1,...,m\); \((iii)\quad | \tau_{\nu,n}| \gg \ll g_{\nu}^{- \lambda_{\nu,n+1}}, \nu =1,...,m\). Then \(f_ 1(g_ 1^{-1}),...,f_ m(g_ m^{-1})\) are algebraically independent.
Note that Theorem 1 has already been generalized by the author [see Acta Math. Sin. 25, 333-339 (1982; Zbl 0494.10022]. From Theorem 1, the author deduces Theorem 2, which is on the algebraic independence of several Mahler’s fractions and is a generalization of Theorem 1 of K. Mahler [Commun. Pure Appl. Math. 29, 717-725 (1976; Zbl 0339.10025)].
Reviewer: K.Yu

MSC:

11J85 Algebraic independence; Gel’fond’s method
11J70 Continued fractions and generalizations
30B10 Power series (including lacunary series) in one complex variable
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