**Numerical Factorization of**

**Multivariate Complex Polynomials**

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

#### Abstract:

One can consider the problem of factoring multivariate
complex polynomials as a special case of the decomposition of a
pure dimensional solution set of a polynomial system into
irreducible components. The importance and nature of this problem
however justify a special treatment. We exploit the reduction to
the univariate root finding problem as a way to sample the
polynomial more efficiently, certify the decomposition with
linear traces, and apply interpolation techniques to construct the
irreducible factors. With a random combination of differentials we
lower multiplicities and reduce to the regular case. Estimates on
the location of the zeroes of the derivative of polynomials
provide bounds on the required precision. We apply our software
to study the singularities of Stewart-Gough platforms.
**2000 Mathematics Subject Classification:**
Primary 13P05, 14A99; Secondary 65H10, 68W30.

**keywords:**
Approximate factorization, divided differences,
generic points, homotopy continuation,
irreducible decomposition, Newton interpolation,
numerical algebraic geometry, monodromy,
multiple roots, polynomial,
Stewart-Gough platform, symbolic-numeric computation,
traces, witness points.

*Theoretical Computer Science* 315(2-3): 651-669, 2004.
Special Issue on Algebraic and Numerical Algorithms
edited by I.Z. Emiris, B. Mourrain, and V.Y. Pan.