PLANE VALUATIONS AND THEIR COMPLETIONS

We give explicit formulae to compute plane valuations. The main tool is a collection of expressions called Hamburger-Noether expansion. These expansions classify valuations and moreover give us some parametric equations for them. We apply the above results to study the completion v of a plane valuation v. To do it we improve the above classification. This improvement affects the valuations associated to formal branches, since they are the only ones presenting differences among the usual invariants associated to v and to v.