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

Abstract 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 Π enables us to detect the connected components of a real algebraic set without critical points for Π. 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.

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