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Algebraic independence by Mahler’s method and \(S\)-unit equations. (English) Zbl 0802.11029
The algebraic independence of Mahler functions and the algebraic independence of their values at suitable algebraic points is discussed. As an example of the results established here we state the following proposition from the text: Let \(f_ r(z)= \sum_{h=0}^ \infty z^{r^ h}\) and \(g_ r(z)= \prod_{h=0}^ \infty (1- z^{r^ h})\), \(r\geq 2\). Let \(\{\omega_ i\}_{i\geq 1}\) be a set of real quadratic irrational numbers such that \(\mathbb{Q} (\omega_ i) \neq \mathbb{Q} (\omega_ j)\) if \(i\neq j\) and put \(F_{\omega_ i} (z)= \sum_{h=1}^ \infty [h\omega_ i] z^ h\). Then for any algebraic number \(\alpha\) with \(0< |\alpha |<1\), \[ f_ r(\alpha) \quad (r\geq 2), \qquad g_ r(\alpha)\quad (r\geq 2), \qquad F_{\omega_ i} (\alpha) \quad (i\geq 1) \] are algebraically independent.
The paper also deals with the case of Mahler functions of several variables. Earlier similar results have been claimed to be true, but their proofs were not completely clear. The correct and clear proof given in the text relies on a clever application of results on the \(S\)-unit equation.

MSC:
11J86 Linear forms in logarithms; Baker’s method
11D99 Diophantine equations
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References:
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