  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
$\widehat{K}$ 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 $V$.
