Individual polynomials in , s = 2 or 3, play an important role in all areas of computational geometry: Their zero sets constitute curves in the plane or surfaces in 3-space, respectively. In Constructive Solid Geometry (CSG), e.g., solids are represented by arithmetic predicates containing linear or polynomial expressions in the three space variables x ₁, x ₂, x ₃, like x ₁ ² + x ₂ ² + x ₃ ² ? 6.25 ? x ₁ ? x ₂ ? x ₃ ? 1.5. The computational handling of such multivariate expressions generally assumes the data to be exact and attempts to retain logical consistency of the results throughout the further computations, like in determining the relative positions of lines or points with respect to other lines or solid bodies.

Polynomial expressions in any number of variables also play a role in the modelling of many nonlinear phenomena in virtually all areas of Scientific Computing. Here, quite generally, the data in the model expressions have a limited, sometimes very low, accuracy. In this chapter, we concern ourselves with individual multivariate polynomials and meaningful computational problems posed for them, under the general premises of approximate data and approximate computation introduced in Chapters 3 and 4.