Burden of henselian valued fields in the Denef-Pas language

TL;DR

This paper investigates the complexity measure called burden of henselian valued fields in the Denef-Pas language, showing it equals the sum of the burdens of the value group and residue field, with implications for NTP2, strength, and finite burden.

Contribution

It establishes that the burden of henselian valued fields with relative quantifier elimination equals the sum of the burdens of their value group and residue field, linking model-theoretic properties.

Findings

Burden of henselian valued fields equals sum of burdens of value group and residue field.

Henselian valued fields are NTP2 if and only if both value group and residue field are.

Henselian valued fields are strong and have finite burden under the same conditions.

Abstract

Motivated by the Ax-Kochen/Ershov principle, a large number of questions about henselian valued fields have been shown to reduce to analogous questions about the value group and residue field. In this paper, we investigate the burden of henselian valued fields in the three-sorted Denef-Pas language. If $T$ is a theory of henselian valued fields admitting relative quantifier elimination (in any characteristic), we show that the burden of $T$ is equal to the sum of the burdens of its value group and residue field. As a consequence, $T$ is NTP${}_2$ if and only if its residue field and value group are; the same is true for the statements "$T$ is strong" and "$T$ has finite burden."