Regular extensions and algebraic relations between values of Mahler functions in positive characteristic

Abstract:

Let $\mathbb {K}$ be a function field of characteristic $p>0$. We have recently established the analogue of a theorem of Ku. Nishioka for linear Mahler systems defined over $\mathbb {K}(z)$. This paper is dedicated to proving the following refinement of this theorem. Let $f_{1}(z),\ldots , f_{n}(z)$ be $d$-Mahler functions such that $\overline {\mathbb {K}}(z)\left (f_{1}(z),\ldots , f_{n}(z)\right )$ is a regular extension over $\overline {\mathbb {K}}(z)$. Then, every homogeneous algebraic relation over $\overline {\mathbb {K}}$ between their values at a regular algebraic point arises as the specialization of a homogeneous algebraic relation over $\overline {\mathbb {K}}(z)$ between these functions themselves. If $\mathbb {K}$ is replaced by a number field, this result is due to B. Adamczewski and C. Faverjon as a consequence of a theorem of P. Philippon. The main difference is that in characteristic zero, every $d$-Mahler extension is regular, whereas in characteristic $p$, non-regular $d$-Mahler extensions do exist. Furthermore, we prove that the regularity of the field extension $\overline {\mathbb {K}}(z)\left (f_{1}(z),\ldots , f_{n}(z)\right )$ is also necessary for our refinement to hold. Besides, we show that when $p\nmid d$, $d$-Mahler extensions over $\overline {\mathbb {K}}(z)$ are always regular. Finally, we describe some consequences of our main result concerning the transcendence of values of $d$-Mahler functions at algebraic points.