Algebraic Independence of Mahler Functions via Radial Asymptotics

We present a new method for algebraic independence results in the context of Mahler’s method. In particular, our method uses the asymptotic behaviour 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 + z2)F (z4) + z4F (z16) = 0. Specifically, we prove that the functions F (z), F (z4), F ′(z), and F ′(z4) are algebraically independent over C(z). An application of a celebrated result of Ku. Nishioka then allows one to replace C(z) by Q when evaluating these functions at a nonzero algebraic number α in the unit disc.