CAD and Topology of Semi-Algebraic Sets

Abstract

Although Cylindrical Algebraic Decomposition (CAD) is widely used to study the topology of semi-algebraic sets (especially algebraic curves), there are very few studies of the topological properties of the output of the CAD algorithms. In this paper three possible bad topological properties of the output of CAD algorithms are described. It is shown that these properties may not occur after a generic change of coordinates and that they may be avoided, in dimension not greater than three, with a modification of the CAD algorithm. As this modification of the CAD algorithm requires to solve some polynomial systems, it is also shown that the computation with real algebraic numbers may be advantageously replaced, in all the variants of the CAD algorithm, by solving systems of polynomial equations and inequalities.