A topological approach to key polynomials

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 K‾ of K with respect to v, a fixed extension of μ|K to 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∘(ai,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).