Abstract Key Polynomials and MacLane-Vaqui\'e chains

TL;DR

This paper explores the relationship between abstract key polynomials and MacLane-Vaquié chains in valued field extensions, providing a theoretical framework for understanding their connection across arbitrary rank valuations.

Contribution

It establishes a new relation between complete sequences of abstract key polynomials and MacLane-Vaquié chains for valued fields of any rank.

Findings

Derived a relation linking key polynomials and MacLane-Vaquié chains

Extended the theory to valued fields of arbitrary rank

Provided a unified framework for valuation extensions

Abstract

In this paper, for a valued field $(K, v)$ of arbitrary rank and an extension $w$ of $v$ to $K(X),$ a relation between induced complete sequences of abstract key polynomials and MacLane-Vaqui\'e chains is given.