Intersections of valuation rings in $K[x, y]$

Junyi Xie (1)

1 IRMAR - Institut de Recherche Mathématique de Rennes

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.

Domains: Algebraic Geometry [math.AG]; Commutative Algebra [math.AC]

Junyi Xie. Intersections of valuation rings in $K[x, y]$. Proceedings of the London Mathematical Society, 2015, 111 (1), pp.240-274. ⟨10.1112/plms/pdv022⟩. ⟨hal-01457895⟩