37 Computational and Quantitative Real Algebraic Geometry

Computational and quantitative real algebraic geometry studies various algorithmic and quantitative questions dealing with the real solutions of a system of equalities, inequalities, and inequations of polynomials over the real numbers. This emerging field is largely motivated by the power and elegance with which it solves a broad and general class of problems arising in robotics, vision, computer-aided design, geometric theorem proving, etc. The algorithmic problems that arise in this context are formulated as decision problems for the first-order theory of reals and the related problems of quantifier elimination (Section 37.1), as well as problems of computing topological invariants of semi-algebraic sets. The associated geometric structures are then examined via an exploration of the semialgebraic sets (Section 37.2). Algorithmic problems for semialgebraic sets are considered next. In particular, Section 37.2 discusses real algebraic numbers and their representation, relying on such classical theorems as Sturm’s theorem and Thom’s Lemma (Section 37.3). This discussion is followed by a description of semialgebraic sets using the concept of cylindrical algebraic decomposition (CAD) in both one and higher dimensions (Sections 37.4 and 37.5). This concept leads to brief descriptions of two algorithmic approaches for the decision and quantifier elimination problems (Section 37.6): namely, Collins’s algorithm based on CAD, and some more recent approaches based on critical points techniques and on reducing the multivariate problem to easier univariate problems. These new approaches rely on the work of several groups of researchers: Grigor’ev and Vorobjov [Gri88, GV88], Canny [Can87, Can90], Heintz et al. [HRS90], Renegar [Ren91, Ren92a, Ren92b, Ren92c], and Basu et al. [BPR96]. In Section 37.7 we describe certain mathematical results on bounding the topological complexity of semi-algebraic sets, and in Section 37.8 we discuss some algorithms for computing topological invariants of semi-algebraic sets. In Section 37.9 we describe some quantitative results from metric semi-algebraic geometry. These have proved useful in applications in computer science. In Section 37.10 we discuss the connection between quantitative bounds on the topology of semi-algebraic sets, and the polynomial partitioning method that have gained prominence recently in discrete and computational geometry. Finally, we give a few representative applications of computational semi-algebraic geometry in Section 37.11.