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Let $\mathbb{K}$ be a function field of characteristic $p>0$. We recentlyestablished the analogue of a theorem of Ku. Nishioka for linear Mahler systemsdefined over $\mathbb{K}(z)$. This paper is dedicated to proving the followingrefinement of this theorem. Let $f_{1}(z),\ldots f_{n}(z)$ be $d$-Mahlerfunctions 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 theirvalues at a regular algebraic point arises as the specialization of ahomogeneous algebraic relation over $\overline{\mathbb{K}}(z)$ between thesefunctions themselves. If $\mathbb{K}$ is replaced by a number field, thisresult is due to B. Adamczewski and C. Faverjon, as a consequence of a theoremof 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 ofthe 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 showthat, when $p\nmid d$, $d$-Mahler extensions over $\overline{\mathbb{K}}(z)$are always regular. Finally, we describe some consequences of our main resultconcerning the transcendence of values of $d$-Mahler functions at algebraicpoints.

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