• DOI:10.1093/imrn/rnv139
• Corpus ID: 31264795

Algebraic independence of Mahler functions via radial asymptotics

@article{Brent2014AlgebraicIO,
  title={Algebraic independence of Mahler functions via radial asymptotics},
  author={Richard P. Brent and Michael Coons and Wadim Zudilin},
  journal={arXiv: Number Theory},
  year={2014}
}

• R. Brent, M. Coons, W. Zudilin
• Published 26 December 2014
• Mathematics
• arXiv: Number Theory

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 $F(z)-(1+z+z^2)F(z^4)+z^4F(z^{16})=0$. 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 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.