Is there a generalization of the fundamental theorem of algebra for power series?

Question

Given the similarity between polynomials and power series, I was wondering if there is any generalization of the fundamental theorem of algebra for power series. I understand that it doesn't make much sense to talk of multiplicity when the roots are supposed to be infinite, but maybe there is something like this :P

Answer 1

One generalization of the fundamental theorem of algebra to entire functions is given by the Little Picard theorem, which can be phrased as follows:

If $f$ is a non-constant entire function and $w\in \mathbb C$, then the equation $f(z)=w$ always has a solution, except perhaps for a single value of $w$.

This statement generalizes the fundamental theorem of algebra, which can be phrased as follows:

If $f$ is a non-constant polynomial function and $w\in \mathbb C$, then the equation $f(z)=w$ always has a solution.