Valuations with infinite limit-depth

TL;DR

This paper constructs a valuation on polynomial rings over a field with an infinite chain of limit augmentations, advancing the understanding of valuation structures in algebra.

Contribution

It introduces a valuation with an infinite Maclane--Vaquié chain, providing new insights into valuation theory and polynomial ring structures.

Findings

Constructed a valuation with an infinite chain of limit augmentations

Demonstrated the existence of such valuations in a specific field setting

Enhanced the theoretical framework of valuation algebra and polynomial rings

Abstract

For a certain field $K$, we construct a valuation-algebraic valuation on the polynomial ring $K[x]$, whose Maclane--Vaqui\'e chain consists of an infinite (countable) number of limit augmentations