Properness Defects of Projections and Computation of at Least One Point in Each Connected Component of a Real Algebraic Set

Computing at least one point in each connected component of a real algebraic set is a basic subroutine to decide emptiness of semi-algebraic sets, which is a fundamental algorithmic problem in effective real algebraic geometry. In this article we propose a new algorithm for the former task, which avoids a hypothesis of properness required in many of the previous methods. We show how studying the set of non-properness of a linear projection enables us to detect the connected components of a real algebraic set without critical points for . Our algorithm is based on this observation and its practical counterpoint, using the triangular representation of algebraic varieties. Our experiments show its efficiency on a family of examples.