Key polynomials and pseudo-convergent sequences

TL;DR

This paper introduces a new concept of key polynomials for valuations on polynomial rings, showing they are irreducible, relate to pseudo-convergent sequences, and can fully determine valuations.

Contribution

It defines a novel type of key polynomials, proves their properties, and establishes their role in characterizing valuations and their connection to pseudo-convergent sequences.

Findings

Key polynomials are irreducible and associated with valuations.

Every valuation admits a sequence of key polynomials that determines it.

Established a relation between 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.