Low dimensional sections of basic semialgebraic sets (Q1328086)

A semi-algebraic set \(S\) in a real algebraic set of \(\mathbb{R}^ n\) is called basic open if there exist a nonnegative integer \(s\) and \(s\) polynomials \(f_ 1, \dots, f_ s\) such that \(S = \{x \in X | f_ 1(x) > 0, \dots, f_ s (x) > 0\}\), and in this case \(S\) is called \(s\)- basic. A basic open semi-algebraic set \(S\) of \(X\) is called generically \(s\)-basic if there exist an algebraic subset \(Z \subset X\) with \(\dim Z < \dim X\) and an \(s\)-basic \(S'\) such that \(S - Z = S' - Z\). By the theorem of Bröcker and Scheiderer, \(S\) is \(s\)-basic if and only if, for every irreducible subset \(Y\) of \(X\), \(S \cap Y\) is generically \(s\)-basic. In this paper the following fundamental result is proved: For a basic open \(S\), \(S\) is \(s\)-basic if and only if, for every irreducible subset \(Y\) of \(X\) with \(\dim Y = s + 1\), \(S \cap Y\) is generically \(s\)-basic. The ingredients of the proof are the theory of fans in spaces of orderings of function fields and the theory of the real spectrum.