Transcendence tests for Mahler functions

We give two tests for transcendence of Mahler functions. For our first, we introduce the notion of the eigenvalue λ_F of a Mahler function F(z) and develop a quick test for the transcendence of F(z) over ℂ(z), which is determined by the value of the eigenvalue λ_F. While our first test is quick and applicable for a large class of functions, our second test, while a bit slower than our first, is universal; it depends on the rank of a certain Hankel matrix determined by the initial coefficients of F(z). We note that these are the first transcendence tests for Mahler functions of arbitrary degree. Several examples and applications are given.