Un principe d'Ax-Kochen-Ershov imaginaire

Abstract

We study interpretable sets in henselian and σ-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 fieldsrelative 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.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.

Elimination of Imaginaries in Ordered Abelian Groups With Bounded Regular Rank

“…The first one is to make the value group as tame as possible (e.g., to assume that it is definably complete) and to understand the obstacles that the residue field naturally contributes to the problem. This research path was successfully finalized by Hils and Rideau-Kikuchi in [9]. 2.…”