Key polynomials in terms of ultrametric balls

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 μK to K. In particular, we show that the ways of augmenting of 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 $K(x)$.