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Algebraic independence measures of the values of Mahler functions. (English) Zbl 0736.11034
Using her own zero estimate of Mahler functions [Acta Arith. 56, No. 3, 249-256 (1990; Zbl 0719.11041)] the author proves fairly good effective algebraic independence measures for the values of those functions:
Theorem 1. Let $${\mathbf A}(z)$$ be an $$m\times m$$ matrix and $${\mathbf B}(z)$$ be an $$m$$-dimensional vector whose entries are rational functions of $$z$$ with algebraic coefficients. Let $${\mathbf F}(z)=^ t(f_ 1(z),\ldots,f_ m(z))$$ be a vector of formal power series with algebraic coefficients which converge in some neighborhood $$U$$ of the point $$z=0$$, satisfying $$(1) {\mathbf F}(z^ d)={\mathbf A}(z){\mathbf F}(z)+{\mathbf B}(z)$$, where $$d\geq2$$ is an integer, and which are algebraically independent over $$\mathbb{C}(z)$$. Suppose that $$\alpha$$ is an algebraic number, $$\alpha\in U$$, $$0<|\alpha|<1$$, and the numbers $$\alpha^{d^ k} (k\geq0)$$ are distinct from the poles of $${\mathbf A}(z)$$ or $${\mathbf B}(z)$$. Then for any $$H$$ and $$s\geq1$$ and for any polynomial $$R\in\mathbb{Z}[x_ 1,\ldots,x_ m{]}$$ whose degree does not exceed $$s$$ and whose coefficients are not greater than $$H$$ in absolute value, the following inequality holds: $| R(f_ 1(\alpha),\ldots,f_ m(\alpha))|>\exp(-\gamma s^ m(\log H+s^{m+2})),$ where $$\gamma$$ is a positive constant depending only on $$\alpha$$ and the functions $$f_ 1,\ldots,f_ m$$. The theorem improves former results of Nesterenko, Miller, Wass, Becker and P. G. Becker and the author [C. R. Math. Acad. Sci., Soc. R. Can. 11, No. 3, 89-93 (1989; Zbl 0685.10026)].

##### MSC:
 11J85 Algebraic independence; Gel’fond’s method 11J91 Transcendence theory of other special functions
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