A numerical method for computing border curves of bi-parametric real polynomial systems and applications

Abstract

For a bi-parametric real polynomial system with parameter values restricted to a finite rectangular region, under certain assumptions, we introduce the notion of border curve. We propose a numerical method to compute the border curve, and provide a numerical error estimation. The border curve enables us to construct a so-called “solution map”. For a given value u of the parameters inside the rectangle but not on the border, the solution map tells the subset that u belongs to together with a connected path from the corresponding sample point w to u. Consequently, all the real solutions of the system at u (which are isolated) can be obtained by tracking a real homotopy starting from all the real roots at w throughout the path. The effectiveness of the proposed method is illustrated by some examples.