Abstract. I will report on a result recently obtained with Christophe Reutenauer.
Let p be a prime number. Mahler’s theorem on interpolation series is a celebrated
result of p-adic analysis. In its simplest form, it states that a function from N
to Z is uniformly continuous for the p-adic metric d_p if and only if it can be
uniformly approximated by polynomial functions. We prove a noncommutative generalization
of this result for functions from a free monoid A* to a free group F(B) (or more
generally to a residually p-finite group), where d_p is replaced by the pro-p metric.
One of the challenges is to find a suitable definition of polynomial functions in this
noncommutative setting.