Key polynomials and pseudo-convergent sequences

Abstract

In this paper we introduce a new concept of key polynomials for a given valuation $\nu$ on $K[x]$. We prove that such polynomials have many of the expected properties of key polynomials as those defined by MacLane and Vaqui\'e, for instance, that they are irreducible and that the truncation of $\nu$ associated to each key polynomial is a valuation. Moreover, we prove that every valuation $\nu$ on $K[x]$ admits a sequence of key polynomials that completely determines $\nu$ (in the sense which we make precise in the paper). We also establish the relation between these key polynomials and pseudo-convergent sequences defined by Kaplansky.