In the numerical treatment of solution sets of polynomial systems, methods for sampling and tracking a path on a solution component are fundamental. For example, in the numerical irreducible decomposition of a solution set for a polynomial system, one first obtains a ``witness point set'' containing generic points on all the irreducible components and then these points are grouped via numerical exploration of the components by path tracking from these points. A numerical difficulty arises when a component has multiplicity greater than one, because then all points on the component are singular. This paper overcomes this difficulty using an embedding of the polynomial system in a family of systems such that in the neighborhood of the original system each point on a higher multiplicity solution component is approached by a cluster of nonsingular points. In the case of the numerical irreducible decomposition, this embedding can be the same embedding that one uses to generate the witness point set. In handling the case of higher multiplicities, this paper, in concert with the methods we previously proposed to decompose reduced solution components, provides a complete algorithm for the numerical irreducible decomposition. The method is applicable to tracking singular paths in other contexts as well.
2000 Mathematics Subject Classification : Primary 65H10, 14Q99; Secondary 68W30.
keywords : Component of solutions, embedding, interpolation, irreducible component, irreducible decomposition, generic point, homotopy continuation, numerical algebraic geometry, multiplicity, path following, polynomial system, sampling, singularity.
In "Algebraic Geometry, a Volume in Memory of Paolo Francia" (ed. by M.C. Beltrametti, F. Catanese, C. Ciliberto, A. Lanteri, C. Pedrini), pages 329-345, W. de Gruyter, 2002.