Non‐archimedean stratifications of tangent cones

Abstract

We study the impact of a kind of non‐archimedean stratifications (t‐stratifications) on tangent cones of definable sets in real closed fields. We prove that such stratifications induce stratifications of the same nature on the tangent cone of a definable set at a fixed point. As a consequence, the archimedean counterpart of a t‐stratification is shown to induce Whitney stratifications on the tangent cones of a semi‐algebraic set. Extensions of these results are proposed for real closed fields with further structure.

Hensel minimality I

“…The existence proof of t-stratifications given in [36] is carried out under some axiomatic assumptions, namely [36,Hypothesis 2.21]. Those assumptions hold in valued fields with analytic structure (in the sense of [14]) by [36,Proposition 5.12] and in power-bounded T-convex structures by [34]. We will now show that the assumptions hold in any 1-h-minimal theory of equi-characteristic 0, hence implying that t-stratifications exist in this generality.…”

“…Note that the proof we gave here also simplifies the proofs from [36] (in the case of fields with analytic structure) and [34] (in the case of T -convex structures): In those papers, the proof of (4) was done using a complicated inductive argument using the existence of t-stratifications in lower dimension. This has been replaced by the more direct proof of our Theorem 5.4.10.…”

“…The proof that Th(K) is 1-h-minimal is essentially contained in the existing literature: Using the criteria given in Theorem 2.5.1, this can be deduced from [34, Theorems 2.1 and 2.9]. However, Theorem 2.9 is a lot deeper than what we really need, so we give a more direct proof below (mainly following the ideas from [34]). 7.2.2.…”

“…Based on those properties, various recent results in the model theory of Henselian valued fields readily generalize to arbitrary Hensel minimal valued fields, like on Lipschitz continuity [9] and t-stratifications (which were introduced in [36] and studied further in [35,34,37]). Moreover, our Taylor approximation results lay the ground for further generalizations of motivic integration (both, the one from [17,18] and the one from [40]) and its applications (e.g.…”

Hensel minimality I

“…A large part of this paper consists of proving our main geometric results in Hensel minimal structures, which are similar to those in o-minimal structures, in particular cell decomposition, dimension theory, the ‘Jacobian Property’ (which plays a key role in constructing motivic integration and can be considered an analogue of the Monotonicity Theorem from the o-minimal context, where is replaced by ), as well as higher-order and higher-dimension versions of the Jacobian Property, which state that definable functions have good approximations by their Taylor polynomials. Based on those properties, various recent results in the model theory of Henselian valued fields readily generalize to arbitrary Hensel minimal valued fields, like those on Lipschitz continuity [11] and t-stratifications (which were introduced in [43] and studied further in [42, 41, 44]). As an extra upshot, Hensel minimality intrinsically has resplendency properties in the spirit of resplendent quantifier elimination: that is, it is preserved by different kinds of expansions of the structure.…”