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Algebraic independence of Mahler functions via radial asymptotics. (English) Zbl 1415.11104
Summary: We present a new method for algebraic independence results in the context of Mahler’s method. In particular, our method uses the asymptotic behavior of a Mahler function \(f(z)\) as \(z\) goes radially to a root of unity to deduce algebraic independence results about the values of \(f(z)\) at algebraic numbers. We apply our method to the canonical example of a degree two Mahler function; that is, we apply it to \(F(z)\), the power series solution to the functional equation \(F(z)-(1+z+z^2)F(z^4)+z^4F(z^{16})=0\). Specifically, we prove that the functions \(F(z)\), \(F(z^4)\), \(F^\prime(z)\), and \(F^\prime(z^4)\) are algebraically independent over \(\mathbb {C}(z)\). An application of a celebrated result of Ku. Nishioka then allows one to replace \(\mathbb {C}(z)\) by \(\mathbb {Q}\) when evaluating these functions at a nonzero algebraic number \(\alpha \) in the unit disc.

11J85 Algebraic independence; Gel’fond’s method
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