Abstract
Many algorithms for determining properties of semi-algebraic sets rely upon the ability to compute smooth points [1]. We present a simple procedure based on computing the critical points of some well-chosen function that guarantees the computation of smooth points in each connected bounded component of a real atomic semi-algebraic set. Our technique is intuitive in principal, performs well on previously difficult examples, and is straightforward to implement using existing numerical algebraic geometry software. The practical efficiency of our approach is demonstrated by solving a conjecture on the number of equilibria of the Kuramoto model for the n = 4 case. We also apply our method to design an efficient algorithm to compute the real dimension of algebraic sets, the original motivation for this research.
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