Characterization of Extremal Valued Fields

TL;DR

This paper characterizes valued fields where the valuation ring's polynomial images always contain an element of maximal value or zero, providing insights into their algebraic structure and valuation properties.

Contribution

It introduces a new characterization of extremal valued fields based on polynomial images of valuation rings, advancing understanding of their valuation-theoretic properties.

Findings

Identifies conditions under which polynomial images contain maximal elements.

Provides a classification of extremal valued fields.

Enhances understanding of valuation ring polynomial behavior.

Abstract

We characterize those valued fields for which the image of the valuation ring under every polynomial in several variables contains an element of maximal value, or zero.