A topological approach to key polynomials

Abstract

In this paper we present characterizations of the sets of key polynomials and abstract key polynomials for a valuation $\hat{I}¼$ of $K(x)$, in terms of (ultrametric) balls in the algebraic closure $\overline{K}$ of $K$ with respect to $v$, a fixed extension of $\hat{I}{¼}_{\mid K}$ to $\overline{K}$. In particular, we show that the ways of augmenting $\hat{I}¼$, in the sense of Mac Lane, are in one-to-one correspondence with the partition of a fixed closed ball $B(a,\hat{I}´)$ associated to $\hat{I}¼$ into the disjoint union of open balls $B^{\circ }(a_i,\hat{I}´)$, 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 $K(x)$.