Semi-algebraic Sets

Abstract

This chapter deals with semi-algebraic sets over a real closed field R. These are the sets defined by a boolean combination of polynomial equations and inequalities. This class of sets has a remarkable property: stability under projection. Several applications of this basic property are investigated. The study of semi-algebraic sets is based mainly on the ∜slicingℝ technique, which makes it possible to decompose them into a finite number of subsets semi-algebraically homeomorphic to open hypercubes. Using this decomposition, we show that a semi-algebraic set has a finite number of semi-algebraically connected components. The notions of connectedness and compactness over a real closed field, other than ℝ, require some care. Nevertheless, closed and bounded semi-algebraic subsets of R ⁿ preserve several of the properties known in the case R = ℝ. They are proved using the curve-selection lemma. All this is the subject of the first five sections of this chapter. In Section 6, we study continuous semi-algebraic functions and we show Łojasiewicz’s inequality. Section 7 deals with the separation of disjoint closed semi-algebraic sets. Section 8 introduces the notion of dimension for a semi-algebraic set and establishes its expected properties. Finally, the last section contains essentially an implicit function theorem in the semi-algebraic framework (this result is well known over R but it is also useful over real closed fields other than ℝ).

Throughout this chapter R is a fixed real closed field.