Intersection of valuation rings in $k[x,y]$

Abstract

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.

References in corpus (2)

• How to determine the sign of a valuation on C[x,y]?
• An effective criterion for algebraic contractibility of rational curves