Definable valuations on ordered fields

TL;DR

This paper investigates the definability of convex valuations, especially henselian valuations, on ordered fields in different logical languages, revealing nuanced differences and conditions for definability.

Contribution

It demonstrates that while some valuations are definable only in the richer language, henselian valuations definable in that language are also definable in the simpler language, with stable embedding results.

Findings

Convex valuations can be definable in ordered ring language but not in ring language.

Any $ ext{L}_{or}$-definable henselian valuation is also $ ext{L}_r$-definable.

In almost real closed fields, all $ ext{L}_{or}$-definable valuations are henselian.

Abstract

We study the definability of convex valuations on ordered fields, with a particular focus on the distinguished subclass of henselian valuations. In the setting of ordered fields, one can consider definability both in the language of rings $\mathcal{L}_{\mathrm{r}}$ and in the richer language of ordered rings $\mathcal{L}_{\mathrm{or}}$. We analyse and compare definability in both languages and show the following contrary results: while there are convex valuations that are definable in the language $\mathcal{L}_{\mathrm{or}}$ but not in the language $\mathcal{L}_{\mathrm{r}}$, any $\mathcal{L}_{\mathrm{or}}$-definable henselian valuation is already $\mathcal{L}_{\mathrm{r}}$-definable. To prove the latter, we show that the value group and the ordered residue field of an ordered henselian valued field are stably embedded (as an ordered abelian group, respectively as an ordered field). Moreover, we show that in almost real closed fields any $\mathcal{L}_{\mathrm{or}}$-definable valuation is henselian.