Un principe d’Ax–Kochen–Ershov imaginaire

Abstract

We study interpretable sets in henselian and $\sigma$-henselian valued fields with value group elementarily equivalent to $\mathbb{Q}$ or $\mathbb{Z}$. 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.