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

TL;DR

This paper refines a theorem on algebraic relations between Mahler functions over function fields of positive characteristic, establishing conditions under which relations among their values are derived from relations among the functions themselves.

Contribution

It proves that regularity of the field extension is necessary and sufficient for algebraic relations of Mahler functions' values to come from relations among the functions, extending prior results to positive characteristic.

Findings

Regularity of the extension is necessary for the main theorem.

In characteristic p, non-regular Mahler extensions exist.

When p does not divide d, all Mahler extensions are regular.

Abstract

Let $\mathbb{K}$ be a function field of characteristic $p>0$. We 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.