Burden of henselian valued fields in the Denef-Pas language

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."