  Given a polynomial system f, a fundamental question is to determine if f has
real roots. Many algorithms involving the use of infinitesimal deformations
have been proposed to answer this question. In this article, we transform an
approach of Rouillier, Roy, and Safey El Din, which is based on a classical
optimization approach of Seidenberg, to develop a homotopy based approach for
computing at least one point on each connected component of a real algebraic
set. Examples are presented demonstrating the effectiveness of this
parallelizable homotopy based approach.
