A measure for semialgebraic sets related to Boolean complexity

Abstract

Let X be a real anne or spherical semialgebraic set over a real closed field R. By spherical set we mean a subset of the manifold of the rays through the origin in the affine space R N+l, or equivalently a subset of RN+I\{O}/R +, defined by homogeneous polynomial relations. It is clear that in this latter space, using equivalence by positive homogeneity, we can define semialgebraic sets by systems of homogeneous polynomial equations and inequalities just as we define projective varieties by systems of homogeneous equations. Spherical geometry enjoys advantages not to be found in affine or projective spaces and in this paper we will show, using natural mappings between spherical and affine spaces, how it can be applied to affine geometry as well. As an application (Proposition 5.1 below) we demonstrate lower bounds for the number of inequalities necessary to define certain affine semialgebraic sets. Our key ingredients are a coarse, highly structured measure for the complexity of affine or spherical semialgebraic sets, a complexity reducing operator for spherical sets, and several complexitymonotonic mappings between spherical and affine spaces. We introduce our complexity measure and an associated filtered algebraic structure for semialgebraic subsets of X in the following sequence of definitions.