On chains associated with abstract key polynomials

TL;DR

This paper explores the relationships between abstract key polynomials, complete sets, chains, and Okutsu frames in henselian valued fields, providing new insights into valuation theory and polynomial extensions.

Contribution

It establishes connections between abstract key polynomials, complete sets, saturated chains, and Okutsu frames, extending valuation theory to arbitrary rank henselian fields.

Findings

Connected abstract key polynomials with Okutsu frames.

Linked complete sets of ABKPs to Maclane-Vaquié chains.

Provided a unified framework for valuation extensions.

Abstract

In this paper, for a henselian valued field $(K, v)$ of arbitrary rank and an extension $w$ of $v$ to $K(X),$ we use abstract key polynomials for $w$ to give a connection between complete sets, saturated distinguished chains and Okutsu frames. Further, for a valued field $(K, v),$ we also obtain a close connection between complete set of ABKPs for $w$ and Maclane-Vaqui\'e chains of $w.$