Definability of henselian valuations by conditions on the value group

TL;DR

This paper investigates the conditions under which henselian valuations are definable in the language of rings, focusing on the role of the value group and establishing the necessity of parameters for definability.

Contribution

It proves that henselian valuations with certain value groups are definable using one parameter and shows this parameter is necessary, strengthening previous results.

Findings

Henselian valuations with non-closed value groups are definable with one parameter.

One parameter is proven to be necessary for such definability.

Introduces a construction method for a t-henselian non-henselian ordered field.

Abstract

Given a henselian valuation, we study its definability (with and without parameters) by examining conditions on the value group. We show that any henselian valuation whose value group is not closed in its divisible hull is definable in the language of rings, using one parameter. Thereby we strengthen known definability results. Moreover, we show that in this case, one parameter is optimal in the sense that one cannot obtain definability without parameters. To this end, we present a construction method for a $t$-henselian non-henselian ordered field elementarily equivalent to a henselian field with a specified value group.