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Non-Archimedean Whitney-stratifications

Immanuel Halupczok,
published 2011-09-27

We define an analogue of Whitney stratifications for Henselian valued fields K of equi-characteristic 0 and prove that such stratifications exist. This analogue is a pretty strong notion; in particular, it sees singularities both at the level of the valued field and of the residue field. Using methods from non-standard analysis, we show how a stratification in our sense can be turned into a classical Whitney stratification of a given (semi-)algebraic subset of \bbR^n or \bbC^n. As in the classical setting, we can work with different classes of subsets of K^n, e.g. algebraic sub-varieties or certain classes of analytic subsets. The general framework are definable sets (in the sense of model theory) in a language which satisfies certain hypotheses. Another point of view is that our result describes sets up to ultra-metric isometry. In a previous article, a conjectural such description has been given for definable subsets of \bbZ_p^n; the present result implies that conjecture when p is sufficiently big.

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