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 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(4)F (z(16))=0. Specifically, we prove that the functions F(z), F(z(4)), F' (z), and F' (z(4)) 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 a in the unit disc.