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

Let K \mathbb {K} be a function field of characteristic p > 0 p>0 . We have recently established the analogue of a theorem of Ku. Nishioka for linear Mahler systems defined over K ( z ) \mathbb {K}(z) . This paper is dedicated to proving the following refinement of this theorem. Let f 1 ( z ) , … , f n ( z ) f_{1}(z),\ldots , f_{n}(z) be d d -Mahler functions such that K ¯ ( z ) ( f 1 ( z ) , … , f n ( z ) ) \overline {\mathbb {K}}(z)\left (f_{1}(z),\ldots , f_{n}(z)\right ) is a regular extension over K ¯ ( z ) \overline {\mathbb {K}}(z) . Then, every homogeneous algebraic relation over K ¯ \overline {\mathbb {K}} between their values at a regular algebraic point arises as the specialization of a homogeneous algebraic relation over K ¯ ( z ) \overline {\mathbb {K}}(z) between these functions themselves. If K \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 d -Mahler extension is regular, whereas in characteristic p p , non-regular d d -Mahler extensions do exist. Furthermore, we prove that the regularity of the field extension K ¯ ( z ) ( f 1 ( z ) , … , f n ( z ) ) \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 ∤ d p\nmid d , d d -Mahler extensions over K ¯ ( z ) \overline {\mathbb {K}}(z) are always regular. Finally, we describe some consequences of our main result concerning the transcendence of values of d d -Mahler functions at algebraic points.