# Singularities of an analytic function over a non-archimedean field

What do we know about the types of singularities that a convergent power series over a non-archimedean field can have?

More specifically:

i) What types of essential singularities can occur?

ii) Are singularities isolated?

iii) Who is known to work on this problem? Edit: The answer is probably "nobody", see Jérôme's comments.

iv) In case the answer to iii) is "nobody": Why not?

Edit: Since I asked the question, I learned the following: Removable singularities are in fact removable (Bartenwerfer 1976).

I also have a new subquestion:

v) Develop an analytic function $f$ into a power series on some polydisc $D$. The function $f$ has a singularity on the boundary $\partial D$ of the disc. Assume the power series we got is an algebraic power series, i.e. an algebraic element over the ring of polynomials $k[T] \subset k[[T]]$. Is it true that $f$ has only finitely many poles and no essential singularities on $\partial D$?

vi) Does v) still hold if we work over a smooth affinoid algebra $A$ whose reduction $\tilde{A}$ is also smooth?

• Hi Helene! In the case of meromorphic functions, I think people mean poles when they write singularities. So in dimension 1, it is indeed a discrete set. Mar 24, 2015 at 13:27
• As for your questions, as you say, it seems to me that the theory is rather trivial and I am not sure that there is much to say. Anyway I did not hear of anybody working on that. Mar 24, 2015 at 13:31
• Hi Jerome! OK, the problem is that you cannot identify the function with the power series, at least not at the boundary of the ball of convergence. Thanks. What do you mean by "rather trivial"? Mar 24, 2015 at 13:44
• When I hear meromorphic function on the line, I understand that it is locally the quotient of two analytic functions around every point of the line. So, in some sense, it is defined everywhere and I do not naturally see a ball of convergence. Do you have an example of a paper where you saw that? Mar 24, 2015 at 13:53
• By "rather trivial", I mean what you said above: the bahaviour of a series is the same everywhere at the "boundary" of the disk of convergence. Mar 24, 2015 at 13:55