A characterization property on fields equivalent to algebraicity on Banach spaces

In 1980, Christol, Kamae, Mendès France and Rauzy stated in [3] an important theorem in automata theory. This theorem links sequences recognized by automata and algebraic formal power series. In 1994, Bruyère, Hansel, Michaux and Villemaire extended this theorem with a logical link in [2]. With theses two articles, we can translate the property for a formal power series to be algebraic in combinatorics terms or logical terms. Our general purpose is to extend these theorems to algebraic dependences between formal power series. We want to be able to translate in combinatorial terms the fact, for two formal power series, to be algebraically dependent. Our first approach, see [4], was combinatorial, and we proved linear independences between some formal power series. The second idea is to use logic (remember that for the case of algebraicity theses points of view are equivalent) and hope it could be translated in combinatorial terms further. That is why we were interested in the work of Tyszka (even if it does not speak about formal power series). Indeed, Tyszka introduce a logical property which is equivalent to algebraicity in R and in the p-adic field Qp. The goal of this article is to study this property and describe fields for which it is equivalent to algebraicity. We will see that the formal power series field is one of them and why finding a good equivalence for algebraic dependence is not easy. Actually, the source of the problem is that we work on a field with positive characteristic so we suggest a property quite different from algebraic dependence but (we hope) more likely equivalent to some combinatorial characterization.