## Power series in P-adic roots of unity

##### Abstract

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.

##### Department

##### Description

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