Homotopies for solving polynomial systems within a bounded domain

Abstract

The problem considered in this paper is the computation of all solutions of a given polynomial system in a bounded domain. Proving Rouche's theorem by homotopy continuation concepts yields a new class of homotopy methods, the so-called regional homotopy methods. These methods rely on isolating a part of the system to be solved, which dominates the rest of the system on the border of the domain. As the dominant part has a sparser structure, it is easier to solve. It will be used as start system in the regional homotopy. The paper further describes practical homotopy construction methods by presenting estimators to obtain bounds for polynomials over a bounded domain. Applications illustrate the usefulness of the approach.