** Numerical Irreducible Decomposition **

** using Projections from Points on the Components **

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

#### Abstract:

To classify positive dimensional solution components of a polynomial
system, we construct polynomials interpolating points sampled from each
component. In previous work, points on an i-dimensional component
were linearly projected onto a generically chosen (i+1)-dimensional
subspace. In this paper, we present two improvements.
First, we reduce the dimensionality of the ambient space by determining
the linear span of the component and restricting to it. Second, if the
dimension of the linear span is greater than i+1, we use a less generic
projection that leads to interpolating polynomials of lower degree, thus
reducing the number of samples needed.
While this more efficient approach still guarantees - with
probability one - the correct determination of the degree of
each component, the mere evaluation of an interpolating polynomial
no longer certifies the membership of a point to that component.
We present an additional numerical test that certifies membership in
this new situation. We illustrate the performance of our new approach
on some well-known test systems.
**2000 AMS Subject Classification :**
Primary 65H10; Secondary 13P05, 14Q99, 68W30.

**keywords :**
Components of solutions, central projection, embedding,
generic points, homotopy continuation, irreducible components,
numerical algebraic geometry, polynomial system, primary decomposition.

In *Symbolic Computation: Solving Equations in Algebra, Geometry, and
Engineering*, edited by E.L. Green, S. Hosten, R. Laubenbacher, and
V.A. Powers. *Contemporary Mathematics*, volume 286, AMS, 2001.