Ubiquity of Lojasiewicz’s example of a nonbasic semialgebraic set.

Abstract

Lojasiewicz pointed out in 1965 that the semialgebraic set {x < 0}[{y < 0} in R2 is not basic: it is not the solution of a simultaneous system of inequalities. In this example the Zariski closure of the topological boundary crosses the set. The purpose of the present paper is to show that this is the only obstruction to a set being basic. A semialgebraic set S contained in a real algebraic set X in Rn is said to be generically basic if there are regular functions f1, . . . , fs, h on X, with h 6= 0, such that S \ {x 2 X | h(x) = 0} = {x 2 X | f1(x) > 0, . . . , fs(x) > 0} \ {x 2 X | h(x) = 0}. Let S be the interior of the closure of Int(S) \ Reg(X). The generic Zariski boundary @ZS of S is defined to be the Zariski closure of Reg(X) \ (S \S). One says that S is crossed by its generic Zariski boundary when (1) dim(S \ @ZS) = d − 1, and (1) S contains some regular points of @ZS of dimension d − 1. If Y is an irreducible algebraic set in Rm and f : Y ! X is a birational map, then the semialgebraic set f−1(S) is called a birational model of S. Theorem: A semialgebraic set S is generically basic if and only if no birational model of S is crossed by its generic Zariski boundary.