On the structure of semialgebraic sets over p-adic fields

Abstract

In his Singular points of complex hypersurfaces Milnor proves the following “curve selection lemma” [10, p. 25]:Let V ⊂ Rm be a real algebraic set, and let U ⊂ Rm 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 curvewith 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 Rn+1 → Rn. If, in this tiny extension of Milnor's result, we replace ‘R’ everywhere by ‘Qp’, 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.