Méthodes fonctionnelles pour la transcendance en caractéristique finie

There are essentially two ways to obtain transcendence results in finite characteristic. The first, historically, is to use Ore's lemma and to prove that a series whose coefficients satisfy well-behaved divisibility properties cannot be a zero of an additive polynomial. This method is of the same kind as the method of p–automata. The second one is to try to imitate the usual methods in characteristic zero and to do transcendence theory with t–modules analogously to what we can do with algebraic groups. We want to show here that transcendence results over Fq(T) can also be obtained with the help of the variable T. If ec(z) is the Carlitz exponential function and e = ec(1), we obtain, in particular, that 1, e, …, e(p–2) (the P–2 first derivative of e with respect to T) are linearly independent over the algebraic closure of Fq(T). A corollary is that for every non-zero element α in Fq((1/T)), αpe and αec(e1/p) are transcendental over Fq(T). By changing the variable and using older results we also obtain the transcendence of ec(ω) for all ω ∈ Fq((1/T)) such that ω(T) and ω(Ti) are not zero and linearly dependent over Fq (Ti) (q > 2i + 1). Such u appear to be transcendental by the method of Mahler if i is not a power of p.