Finding one point on each semi-algebraically connected component of a real algebraic variety, or at least deciding if such a variety is empty or not, is a fundamental problem of computational real algebraic geometry. Although numerous studies have been done on the subject, only a small number of efficient implementations exist.
In this paper, we propose a new efficient and practical algorithm for computing such points. By studying the critical points of the restriction to the variety of the distance function to one well chosen point, we show how to provide a set of zerodimensional systems whose zeros contain at least one point on each semi-algebraically connected component of the studied variety, without any assumption either on the variety (smoothness or compactness for example) or on the system of equations which define it.
From the output of our algorithm, one can then apply, for each computed zerodimensional system, any symbolic or numerical algorithm for counting or approximating the real solutions. We report some experiments using a set of pure exact methods. The practical efficiency of our method is due to the fact that we do not apply any infinitesimal deformations, unlike the existing methods based on a similar strategy.