Algebraic Independence of Mahler Functions via Radial Asymptotics

Abstract

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 |$F(z)$|⁠, |$F(z^4)$|⁠, |$F'(z)$|⁠, and |$F'(z^4)$| 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.Communicated by Prof. Umberto Zannier