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Transcendence measures and nonlinear functional equations of Mahler type. (English) Zbl 0704.11014

Let f be holomorphic in \(| z| <1\) and satisfy a functional equation \(f(z^ p)=Q_ 1(z,f(z))/Q_ 2(z,f(z))\) with \(p\in {\mathbb{N}}\), \(p\geq 2\), \(Q_ i\in K[z,y]\) \((i=1,2)\), where K denotes an algebraic number field, and let \(\alpha\) be algebraic with \(Q_ 2(\alpha^{p^ k},f(\alpha^{p^ k}))\neq 0\) (k\(\in {\mathbb{N}})\). In special cases of the functional equation Galochkin, Miller, Becker-Landeck and Molchanov proved transcendence measures for the values f(\(\alpha\)), which depend on different bounds for the Taylor coefficients of f. In this paper a general zero order estimate and an inequality are proved, from which by the different bounds for the Taylor coefficients the results mentioned above and more general transcendence measures, especially in the case \(p\leq \deg_ yQ_ i<p^ 2(i=1,2)\), can easily be deduced.
Reviewer: T.Töpfer

MSC:

11J82 Measures of irrationality and of transcendence
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References:

[1] P.-G.Becker-Landeck, Quantitative Resultate im Zusammenhang mit der Mahlerschen Transzendenzmethode. Dissertation, Univ. K?ln 1984.
[2] P.-G. Becker-Landeck, Transcendence measures by Mahler’s transcendence method. Bull. Austral. Math. Soc.33, 59-65 (1986). · Zbl 0571.10033 · doi:10.1017/S0004972700002860
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