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Preprint Number 394

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394. Immanuel Halupczok
Non-Archimedean Whitney-stratifications
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Submission date: 27 September 2011.

Abstract:

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 R^n or C^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 Z_p^n; the present result implies that conjecture when p is sufficiently big.

Mathematics Subject Classification: 12J25, 03C60, 03H05, 14B05, 14B20, 32S60

Keywords and phrases:

Full text arXiv 1109.5886: pdf, ps.


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