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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)].

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