Homotopy continuation methods are numerically stable algorithms to approximate all isolated solutions of polynomial systems. These methods provide constructive proofs for Bezout's theorem, are able to exploit sparsity, and provide effective algorithms for a numerical Schubert calculus [1].
A recent development [2] applies homotopies to find representations of all irreducible components of the solution sets, for all dimensions and degrees. For this problem, we employ homotopies in three ways: 1) to compute generic points on every component, for all dimensions; 2) to determine whether a given point belongs to a certain component; 3) to factor solution sets into irreducible components. Applications from mechanism design [3] motivate the development and use of homotopy algorithms.
[1] Jan Verschelde: "Algorithm 795: PHCpack: A general-purpose solver for polynomial systems by homotopy continuation". ACM Transactions on Mathematical Software 25(2): 251-276, 1999.
[2] Andrew J. Sommese, Jan Verschelde, and Charles W. Wampler: "Numerical Irreducible Decomposition using PHCpack". To appear in "Mathematics and Visualization" (ed. by M. Joswig and N. Takayama).
[3] Andrew J. Sommese, Jan Verschelde, and Charles W. Wampler: "Advances in Polynomial Continuation for Solving Problems in Kinematics". Accepted by the Mechanisms and Robotics sub-conference of ASME DETC2002 (Design Engineering Technical Conferences).
CBMS Lectures Series at Texas A&M University on "Solving systems of polynomial equations" by Bernd Sturmfels; May 20-24, 2002.