Approximation of subanalytic sets by normal cones

Abstract

Let A be a closed semialgebraic subset of Euclidean space of codimension at least one, and containing the origin O as a non-isolated point. We prove that, for every real s ≥ 1, there exists an algebraic set V which approximates A to order s at O. The special case s = 1 generalizes the result of the authors that every semialgebraic cone of codimension at least one is the tangent cone of an algebraic set.