Local Approximation of Semialgebraic Sets and Tangent Cones

Tangent cones Let A ⊆ R be a closed semialgebraic (s.a.) set and assume from now on that the origin O is not an isolated point of A. We recall that the tangent cone to A at O is the set C(A) = {u ∈ R|u = lim k→∞ tkxk}, where {xk} ⊂ A tends to O and {tk} ⊂ R. If we allow tk ∈ R only, we get what we call the tangent semicone to A at O, which we denote with C(A). It is easy to show that both C(A) and C(A) are closed s.a. sets with dimension ≤ dim(A) in O, but it is not in general true that they are algebraic if A is algebraic. For instance