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 $\hat{K}$ 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 $V$.

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References in corpus (1)

• The Zariski-Riemann space of valuation domains associated to pseudo-convergent sequences

Cited by in corpus (5)

• The Zariski-Riemann space of valuation domains associated to pseudo-convergent sequences
• Stacked Pseudo-Convergent Sequences and Polynomial Dedekind Domains
• Valuations on $K[x]$ approaching a fixed irreducible polynomial
• Nontriviality of rings of integral-valued polynomials
• Metrizability of spaces of valuation domains associated to pseudo-convergent sequences