Homotopies for solving polynomial systems within a bounded domain (Q1338223)

The authors prove a multidimensional version of Rouché's theorem by homotopy continuation concepts. This result is then applied to give homotopy methods for calculating all of the zeros of a given system of polynomial equations within a bounded domain \(D\). The methods rely on isolating a part of the system which dominates the rest on the boundary \(\partial D\). In general, the dominant part can be so chosen that it has a sparser structure and hence is easier to solve. This can be used as the starting map for a homotopy method for the restriction to \(D\). The paper describes some practical homotopy constructions and gives some numerical examples.