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 $K\left(x\right)$, in terms of (ultrametric) balls in the algebraic closure $\bar{K}$ of K with respect to v, a fixed extension of ${\mu }_{|K}$ to $\bar{K}$. 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 $B(a,\delta )$ associated to μ into the disjoint union of open balls $B^{\circ }(a_i,\delta )$, 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 $K\left(x\right)$.