## 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.

### MSC:

 11J85 Algebraic independence; Gel’fond’s method

### Keywords:

Mahler’s method; algebraic independence; Mahler functions
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