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

Abstract

Let $V$ be a valuation domain with quotient field $K$. Given a pseudo-convergent sequence $E$ in $K$, we study two constructions associating to $E$ a valuation domain of $K(X)$ lying over $V$, especially when $V$ has rank one. The first one has been introduced by Ostrowski, the second one more recently by Loper and Werner. We describe the main properties of these valuation domains, and we give a notion of equivalence on the set of pseudo-convergent sequences of $K$ characterizing when the associated valuation domains are equal. Then, we analyze the topological properties of the Zariski-Riemann spaces formed by these valuation domains.

any comment is welcome! Trans. Amer. Math. Soc. 373 (2020), no. 11, 7959-7990

References in corpus (1)

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

Cited by in corpus (4)

• Extending valuations to the field of rational functions using pseudo-monotone sequences
• Stacked Pseudo-Convergent Sequences and Polynomial Dedekind Domains
• Valuations on $K[x]$ approaching a fixed irreducible polynomial
• Metrizability of spaces of valuation domains associated to pseudo-convergent sequences