Gwladys Fernandes

Let $\mathbb{K}$ be a field and let $d\geq 2$ be an integer. We say that a power series $f(z)\in \mathbb{K}[[z]]$ is a $d$-Mahler function over $\mathbb{K}(z)$ if there exist polynomials $P_{0}(z),\ldots, P_{n}(z)\in\mathbb{K}[z]$, $P_{n}(z)$ $\not\equiv$  $0$, such that \[P_{0}(z)f(z)+P_{1}(z)f(z^{d})+\cdots+P_{n}(z)f(z^{d^{n}})=0.\]

Let $f_{1}(z),\ldots,f_{n}(z)$ be $d$-Mahler functions and let $\alpha$ be an algebraic number over $\mathbb{K}$. We present and illustrate in this talk the following result. Under some assumptions we will make explicit, every algebraic homogeneous relation over $\overline{\mathbb{K}}$ between $f_{1}(\alpha),\ldots,f_{n}(\alpha)$ arises as the specialisation at $z=\alpha$ of a homogeneous algebraic relation over $\overline{\mathbb{K}}(z)$ between $f_{1}(z),\ldots,f_{n}(z)$. When $\mathbb{K}$ is a number field, this result is due to B. Adamczewski and C. Faverjon, as a consequence of a result of P. Philippon.