Extending valuations to the field of rational functions using pseudo-monotone sequences

Abstract

Let V be a valuation domain with quotient field K. We show how to describe all extensions of V to K(X) when the V-adic completion Kˆ is algebraically closed, generalizing a similar result obtained by Ostrowski in the case of one-dimensional valuation domains. This is accomplished by realizing such extensions by means of pseudo-monotone sequences, a generalization of pseudo-convergent sequences introduced by Chabert. We also show that the valuation rings associated to pseudo-convergent and pseudo-divergent sequences (two classes of pseudo-monotone sequences) roughly correspond, respectively, to the closed and the open balls of K in the topology induced by V.