**A Method for Tracking Singular Paths**
**with Application to**
**the Numerical Irreducible Decomposition**

*Andrew J. Sommese*,
*Jan Verschelde*, and
*Charles W. Wampler*

#### Abstract:

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.