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On the structure of semialgebraic sets over p-adic fields

Published online by Cambridge University Press:  12 March 2014

Philip Scowcroft  and
Lou van den Dries
Philip Scowcroft
Affiliation:
Department of Mathematics, Yale University, New Haven, Connecticut 06520
Lou van den Dries*
Affiliation:
Department of Mathematics, Stanford University, Stanford, California 94305
*
Department of Mathematics, University of Illinois, Urbana, Illinois 61801
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In his Singular points of complex hypersurfaces Milnor proves the following “curve selection lemma” [10, p. 25]:

Let V ⊂ R m be a real algebraic set, and let U ⊂ R m be an open set defined by finitely many polynomial inequalities:

Lemma 3.1. If U ∩ V contains points arbitrarily close to the origin (that is if 0 ∈ Closure (U ∩ V)) then there exists a real analytic curve

with p(0) = 0 and with p(t) ∈ U ∩ V for t > 0.

Of course, this result will also apply to semialgebraic sets (finite unions of sets U ∩ V), and by Tarski's theorem such sets are exactly the sets obtained from real varieties by means of the finite Boolean operations and the projection maps R n+1 → R n . If, in this tiny extension of Milnor's result, we replace ‘R’ everywhere by ‘Q p ’, we obtain a p-adic curve selection lemma, a version of which we will prove in this essay. Semialgebraic sets, in the p-adic context, may be defined just as they are over the reals: namely, as those sets obtained from p-adic varieties by means of the finite Boolean operations and the projection maps . Analytic maps are maps whose coordinate functions are given locally by convergent power series.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 53 , Issue 4 , December 1988 , pp. 1138 - 1164
Copyright
Copyright © Association for Symbolic Logic 1988

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References

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