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

Gwladys Fernandes (1)

1 CTN - Combinatoire, théorie des nombres

Let 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 K(z). This paper is dedicated to proving the following refinement of this theorem. Let f 1 (z),. .. fn(z) be d-Mahler functions such that K(z) (f 1 (z),. .. , fn(z)) is a regular extension over K(z). Then, every homogeneous algebraic relation over K between their values at a regular algebraic point arises as the specialization of a homogeneous algebraic relation over K(z) between these functions themselves. If 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 K(z) (f 1 (z),. .. , fn(z)) is also necessary for our refinement to hold. Besides, we show that, when p ∤ d, d-Mahler extensions over 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.

Domaines: Théorie des nombres [math.NT]

Gwladys Fernandes. Regular extensions and algebraic relations between values of Mahler functions in positive characteristic. Transactions of the American Mathematical Society, In press, ⟨10.1090/tran/7798⟩. ⟨hal-01852496⟩