Arithmetic Geometry and Commutative Algebramathscidoc:1908.07001
Proc. London Math. Soc. , 2015
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.