Thanks. Is there any notion of singularity or complete analytic function at all? For example one could take the the polynomial $y^2 - x$ and (as far as I can tell) use Hensel's lemma to expand a power series $p(x) = \sum c_i (x-\alpha)^i$ around every point $\alpha \in \mathbb{C}_p - \{0\}$, in fact, a pair of power series, such that $(x,p(x))$ is identically zero on $y^2 - x$ It seems reasonable to assume that such power series bear some relationship to each other, that an arbitrary pair of power series do not.

Thanks, I didn't think about exchangeability when I considered the problem.

Thanks, I think that set of notes is exactly the sort of thing I was looking for.