Real Algebra

In this chapter, we focus our attention on real algebra and real geometry. We deal with algebraic problems with a formulation over real numbers, ℝ (or more generally, over real closed fields). The underlying (real) geometry provides a rich set of mechanisms to describe such topological notions as “between,” “above/below,” “internal/external,” since it can use the inherent order relation (<) of the real (or, real closed) field. As a result, the subject has found many applications in such practical areas as computer-aided manufacturing, computational geometry, computer vision, geometric theorem proving, robotics, and solid modeling, etc., and thus has generated a renewed interest. We concentrate on the following key ingredients of real algebra: Sturm theory, algebraic numbers, and semialgebraic geometry.