Solving systems of polynomial equations is an important problem in mathematics with a wide range of applications in many fields. The homotopy continuation method is a large class of reliable and efficient numerical methods for solving systems of polynomial equations. An essential component in the homotopy continuation method is the path tracking algorithm for tracking smooth paths of one real dimension. ``Divergent paths'' pose a tough challenge to path tracking algorithms as the tracking of such paths are generally impossible. The existence of such paths is, in part, caused by C^n, the space in which homotopy methods usually operate, being non-compact. A well known remedy is to operate inside the complex projective space instead. Path tracking inside the complex projective space is the focus of this article. An existing method based on the use of generic ``affine charts'' of the complex projective space is widely used. While it works well in theory, we will point out the unnecessary numerical instability it could potentially create via the analysis of ``path condition''. This article, then proposes a numerically superior approach for projective path tracking developed from the point of view of the Riemannian geometry of the complex projective space.
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