Perturbed homotopies for finding all isolated solutions of polynomial systems

Given a polynomial system f:CN→Cn, the methods of numerical algebraic geometry produce numerical approximations of the isolated solutions of f(z)=0, as well as points on any positive-dimensional components of the solution set, V(f). Some of these methods are guaranteed to find all isolated solutions (nonsingular and singular alike), while others may not find singular solutions. One of the most recent advances in this field is regeneration, an equation-by-equation solver that is often more efficient than other methods. However, the basic form of regeneration will not necessarily find all isolated singular solutions of a polynomial system without the additional cost of using deflation.The aim of this article is two-fold. First, more generally, we consider the use of perturbed homotopies for solving polynomial systems. In particular, we propose first solving a perturbed version of the polynomial system, followed by a parameter homotopy to remove the perturbation. Such perturbed homotopies are sometimes more efficient than regular homotopies, though they can also be less efficient. Second, a useful consequence is that the application of this perturbation to regeneration will yield all isolated solutions, including all singular isolated solutions. This version of regeneration - perturbed regeneration - can decrease the efficiency of regeneration but increases its applicability.