Key polynomials and minimal pairs

Abstract

In this paper we establish the relation between key polynomials (as defined in \cite{SopivNova}) and minimal pairs of definition of a valuation. We also discuss truncations of valuations on a polynomial ring $K[x]$. We prove that a valuation $\nu$ is equal to its truncation on some polynomial if and only if $\nu$ is valuation-transcendental. Another important result of this paper is that if $\mu$ is any extension of $\nu$ to $\overline{K}[x]$ and $\Lambda$ is a complete sequence of key polynomials for $\nu$, without last element, then for each $Q\in \Lambda$ there exists a suitable root $a_Q\in \overline{K}$ of $Q$ such that $\{a_Q{\}}_{Q\in \Lambda }$ is a pseudo-convergent sequence defining $\mu$.