Power series in P-adic roots of unity

Motivated by [5], we develop an analogy with a similar problem in p-adic power series over a finite field extension of Qp, say K. Concerned with the convergence of the p-adic power series, we naturally assume that it converges in the unit disc, since we calculate the values of this power series at roots of unity in Q¯ p. This dissertation is devoted to the proof of the following result. Let F(x1,...,xn) be a power series over K, a finite field extension of Qp, converging in On K = {(x1,...,xn) ∈ Kn| max 1≤i≤n {|xi|p} ≤ 1}. Then, there exists a positive constant c such that for any roots of unity ζ1,...,ζn in the algebraic closure of Qp either F(ζ1,...,ζn) = 0 or |F(ζ1,...,ζn)|p ≥ c. We also compute some constants c associated to certain power series as illustrations of the result. In these examples, we realize that the constant c is not unique nor does it follow a pattern. Unfortunately, it’s not known any general formula for c.