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.

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