A topological approach to key polynomials
Enric Nart, Josnei Novacoski, Giulio Peruginelli
TL;DR
This work connects valuation theory on $K(x)$ with ultrametric balls in the algebraic closure of $K$ under a fixed base valuation. By encoding extensions, augmentations, and limits through balls $B(a,\delta)$ and their partitions, it provides ball-based descriptions of key polynomials ${\rm KP}(\mu)$ and abstract key polynomials $\Psi(\mu)$, and characterizes limit key polynomials ${\rm KP}_\infty(\mathcal C)$ via intersections of balls. A central contribution is the explicit correspondence between Mac Lane–Vaquié augmentation steps and ball partitions modulo the decomposition group, yielding a geometric, combinatorial view of the valuation tree and lifting properties across $\overline K(x)$. The results unify several strands (Mac Lane, Vaquié, Nart–Josnei) and give constructive procedures to obtain complete abstract KP sets and Mac Lane–Vaquié chains directly from ultrametric balls. Overall, the paper advances a topological and metric perspective on the classical valuation-theoretic construction of extensions of $v$ to $K(x)$, with clear implications for building and analyzing valuation chains and their limit behavior.
Abstract
In this paper we present characterizations of the sets of key polynomials and abstract key polynomials for a valuation $μ$ of $K(x)$, in terms of (ultrametric) balls in the algebraic closure $\overline K$ of $K$ with respect to $v$, a fixed extension of $μ_{\mid K}$ to $\overline K$. In particular, we show that the ways of augmenting $μ$, in the sense of Mac Lane, are in one-to-one correspondence with the partition of a fixed closed ball $B(a,δ)$ associated to $μ$ into the disjoint union of open balls $B^\circ(a_i,δ)$, modulo the action of the decomposition group of $v$. We also present a similar characterization for the set of limit key polynomials for an increasing family of valuations of $K(x)$.