×

Classification of obstructions for separation of semialgebraic sets in dimension 3. (English) Zbl 0918.14021

Let \(M\) be an irreducible and compact non-singular real algebraic variety and let \(A,B\) be two open semi-algebraic subsets of \(M\). We say that \(A\) and \(B\) are separable, if there exists a polynomial function defined on \(M\), such that for \(Y\) being the Zariski closure of \(\overline A\cap \overline B\), the function \(f\) maps \(A\setminus Y\) only in positive and \(B\setminus Y\) only in negative values.
The first result of the paper is theorem 2.1, where the authors give an abstract algebraic criterion for the separability of the sets \(A\) and \(B\). This criterion is formulated in terms of certain suitable finite sets of orderings of the rational function field of the irreducible real algebraic variety \(M\) (these finite set of orderings are called geometric spaces of orderings). Then, in theorem 2.4, this criterion is restated in more geometric terms (of “walls”, “shadows” and “counter-shadows”). Finally the authors conclude that the separability of \(A\) and \(B\) can be effectively decided if all the ingredients of the problem, namely the sets \(M\), \(A\) and \(B\), are given by polynomial equalities and inequalities. The paper ends with a complete description of all obstructions (in terms of geometric spaces of orderings) for the separability of \(A\) and \(B\) in the case that \(M\) has dimension 3.

MSC:

14P10 Semialgebraic sets and related spaces
PDFBibTeX XMLCite
Full Text: EuDML