On the structure of semialgebraic sets over p-adic fields

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.