Un principe d'Ax-Kochen-Ershov imaginaire

TL;DR

This paper extends Ax-Kochen-Ershov principles to interpretability and imaginaries in henselian valued fields, including difference fields, with implications for model theory and algebraic structures.

Contribution

It introduces an Ax-Kochen-Ershov type principle for weak elimination of imaginaries in henselian fields and extends it to valued difference fields, including non-standard Frobenius automorphisms.

Findings

Weak elimination of imaginaries in henselian fields relative to value group and residual imaginaries.

Elimination of imaginaries in geometric sorts for existentially closed difference valued fields.

Auxiliary results on separated pairs of henselian fields and imaginaries in linear structures.

Abstract

We study interpretable sets in henselian and sigma-henselian valued fields with value group elementarily equivalent to Q or Z. Our first result is an Ax-Kochen-Ershov type principle for weak elimination of imaginaries in finitely ramified characteristic zero henselian fields -- relative to value group imaginaries and residual linear imaginaries. We extend this result to the valued difference context and show, in particular, that existentially closed equicharacteristic zero multiplicative difference valued fields eliminate imaginaries in the geometric sorts; the omega-increasing case corresponds to the theory of the non-standard Frobenius automorphism acting on an algebraically closed valued field. On the way, we establish some auxiliary results on separated pairs of characteristic zero henselian fields and on imaginaries in linear structures which are also of independent interest.