Key polynomials and minimal pairs

TL;DR

This paper explores the relationship between key polynomials and minimal pairs in valuation theory, establishing conditions for valuation truncations and linking key polynomial sequences to pseudo-convergent sequences.

Contribution

It introduces new connections between key polynomials, minimal pairs, and valuation truncations, advancing understanding of valuation extensions and their polynomial representations.

Findings

Valuation equals its truncation on a polynomial iff it is valuation-transcendental.

Existence of roots for key polynomials forming pseudo-convergent sequences.

Characterization of valuation extensions via key polynomial sequences.

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 $\mu$.