Smooth points on semi-algebraic sets

Many algorithms for determining properties of real algebraic or semi-algebraic sets rely upon the ability to compute smooth points. In this paper, we present a procedure based on computing critical points some well-chosen function that guarantees computation in each bounded connected component (real) atomic set. Our technique is intuitive principal, performs well previously difficult examples, and straightforward implement using existing numerical geometry software. The practical efficiency our approach demonstrated by solving conjecture number equilibria Kuramoto model n=4 case. We also apply method design an algorithm dimension sets, original motivation research. compare methods benchmark family.