Non-archimedean stratifications in T-convex fields.

We prove that whenever T is a power-bounded o-minimal theory, t-stratifications exist for definable maps and sets in T-convex fields. To this effect, a thorough analysis of definability in T-convex fields is carried out. One of the conditions required for the result above is the Jacobian property, whose proof in this work is a long and technical argument based on an earlier proof of this property for valued fields with analytic structure. An example is given to illustrate that t-stratifications do not exist in general when T is not power-bounded. We also show that if T is power-bounded, the theory of all T-convex fields is b-minimal with centres.

We also address several applications of tstratifications. For this we exclusively work with a power-bounded T. The first application establishes that a t-stratification of a definable set X in a T-convex field induces t stratifications on the tangent cones of X. This is a contribution to local geometry and singularity theory. Regarding R as a model of T, the remaining applications are derived by considering the stratifications induced on R by t-stratifications in non-standard models. We prove that each such induced stratification is a C1-Whitney stratification; this in turn leads to a new proof of the existence of Whitney stratifications for definable sets in R. We also deal with interactions between tangent cones of definable sets in R and stratifications.