We present a framework for tame geometry on Henselian valued fields, which we call Hensel minimality. In the spirit of o-minimality, which is key to real geometry and several diophantine applications, we develop geometric results and applications for Hensel minimal structures that were previously known only under stronger, less axiomatic assumptions. We show the existence of t-stratifications in Hensel minimal structures and Taylor approximation results that are key to non-Archimedean versions of Pila–Wilkie point counting, Yomdin’s parameterization results and motivic integration. In this first paper, we work in equi-characteristic zero; in the sequel paper, we develop the mixed characteristic case and a diophantine application.

We introduce a class of theories called metastable, including the theory of algebraically closed valued fields (ACVF) as a motivating example. The key local notion is that of definable types dominated by their stable part. A theory is metastable (over a sort Γ) if every type over a sufficiently rich base structure can be viewed as part of a Γ-parametrized family of stably dominated types. We initiate a study of definable groups in metastable theories of finite rank. Groups with a stably dominated generic type are shown to have a canonical stable quotient. Abelian groups are shown to be decomposable into a part coming from Γ, and a definable direct limit system of groups with stably dominated generic. In the case of ACVF, among definable subgroups of affine algebraic groups, we characterize the groups with stably dominated generics in terms of group schemes over the valuation ring. Finally, we classify all fields definable in ACVF.

In this paper, we give a general criterion for elimination of imaginaries using an abstract independence relation. We also study germs of definable functions at certain well-behaved invariant types. Finally we apply these results to prove the elimination of imaginaries in bounded pseudo p-adically closed fields.so-called density theorems [Mon17b, Theorem 3.17 and 6.11] -which generalizes cellular decomposition to the multi-valued (respectively multi-ordered) setting. She also proves an amalgamation theorem [Mon17b, Theorem 3.21 and 6.13] which is a weaker version of the independence theorem of simple theories which holds in this non-simple setting. She then uses those new tools to prove new classification results, the main one [Mon17b, Theorem 4.24 and 8.5] being that bounded pseudo p-adically closed and pseudo real closed fields are NTP 2 of finite burden. One particularity that stands out in the first author's work is that, although most results are proved for both classes, elimination of imaginaries was only proved for pseudo real closed fields [Mon17a]. The initial motivation for this paper was to repair that asymmetry. As it turns out, we were also able to repair a small gap in the proof of the pseudo real closed case, cf. §2.8.1. Since bounded pseudo p-adically closed fields that are not pseudo algebraically closed fields come with finitely many definable valuations, one cannot expect elimination of imaginaries in a language with just one sort for the field, contrary to what happens with pseudo real closed fields. But since the work of [HHM06], we know how to circumvent that particular issue: we have to add, for each valuation, codes for certain definable modules over the valuation ring; that is, work in the so-called geometric language. The main question regarding the imaginaries in bounded pseudo p-adically closed fields then becomes to prove that there are no imaginaries arising from the interaction between the various valuations and, therefore, that it suffices to add the geometric sorts for each of the valuations. The main result of this paper, Theorem (2.58), is a positive answer to this question. We deduce it from our abstract criterion, Proposition (1.17), applied to quantifier free invariant independence, cf Definition (1.18). The results of Section 1.2 play a fundamental role by allowing us to deduce n-ary extension from unary extension for that particular independence relation. Our first step towards this elimination result, and a core ingredient of the rest of the paper, is Proposition (2.26) which states that not only are the geometric sorts for each valuation orthogonal but also that the structure of any given geometric sort is the one induced by the relevant p-adic closure. We then proceed to deduce, from this strong statement on the independence of the valuations, a result on the structure of definable subsets of the valued field, where, at first sight, the valuations do interact. The key ingredient of the first author's proof that pseudo real closed fields eliminate imaginaries is [Mon17b, ...

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.