On the ranks and implicit constant fields of valuations induced by pseudo monotone sequences

TL;DR

This paper investigates valuations induced by pseudo monotone sequences in valued fields, characterizing their implicit constant fields, conditions for their realization, and the possible ranks of these valuations, especially when the base valuation has finite rank.

Contribution

It provides a detailed analysis of the implicit constant fields of valuations induced by pseudo monotone sequences and characterizes when such valuations are value transcendental extensions.

Findings

Implicit constant field is the henselization of (K,v).

Conditions identified for a valuation to be induced by a pseudo monotone sequence.

Complete description of possible ranks of v_E when v has finite rank.

Abstract

Given a valued field $(K,v)$ and a pseudo monotone sequence $E$ in $(K,v)$, one has an induced valuation $v_E$ extending $v$ to $K(X)$. After fixing an extension of $v_E$ to a fixed algebraic closure $\overline{K(X)}$ of $K(X)$, we show that the implicit constant field of the extension $(K(X)|K,v_E)$ is simply the henselization of $(K,v)$. We consider the question: given a value transcendental extension $w$ of $v$ to $K(X)$ and a pseudo monotone sequence $E$ in $(K,v)$, under which precise conditions is $w$ induced by $E$? The dual nature of pseudo convergent sequences of algebraic type and pseudo divergent sequences is also explored. Further, we provide a complete description of the various possibilities of the rank of the valuation $v_E$, provided that $v$ has finite rank.