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