Intersections of valuation rings in $K[x, y]$
Résumé
We associate to any given finite set of valuations on the polynomial ring in two variables overan algebraically closed field a numerical invariant whose positivity characterizes the case whenthe intersection of their valuation rings has maximal transcendence degree over the base field.As an application, we give a criterion for when an analytic branch at infinity in the affineplane that is defined over a number field is, in a suitable sense, the branch of an algebraic curve.