Neighborhoods of Algebraic Sets

Abstract

Abstract. In differential topology, a smooth submanifold in a manifold has a tubular neighborhood, and in piecewise-linear topology, a subcomplex of a simplicial complex has a regular neighborhood. The purpose of this paper is to develop a similar theory for algebraic and semialgebraic sets. The neighborhoods will be defined as level sets of polynomial or semialgebraic functions.

Semi-algebraic neighborhoods of closed semi-algebraic sets

“…This fact is proved in the same way as Lemma 1.8 in [8]. Hence there exists δ such that ∇f 1 and ∇f 2 are nonzero and do not point in opposite direction on f…”

“…Proof. -If X is compact, this is already proved by Durfee [8] and Lojaziewicz [19], [20]. So let us assume that X is not compact.…”

“…In this section, we define the notion of a ρ-quasiregular approaching function for a closed semi-algebraic set, which generalizes the notion of a rug function introduced by Durfee [8].…”

“…Let us compare our notion of ρ-quasiregular approaching function with the notion of rug function due to Durfee [8]. If X is a compact algebraic set of R n , a rug function for X is a proper nonnegative polynomial f such that X = f −1 (0).…”

“…This function f may not be proper as it is explained by the following example due to H. King: let f (x, y) = (x 2 + y 2 ) y(x 2 + 1) − 1 2 + y 2 , then f −1 (0) = {0} but f (x, (1 + x 2 ) −1 ) → 0 as |x| → +∞. Durfee [8] proved that any compact algebraic set X can be written as the set of zeros of a proper nonnegative polynomial function g. Following Thom's terminology, he called such a function a rug function for X. Then he defined the notion of algebraic neighborhood: a subset T with X ⊂ T ⊂ R values of g. Using the gradient vector field of g, he showed that the inclusion X ⊂ T is a homotopy equivalence.…”