Joint with Andrew J. Sommese and Charles W. Wampler we proposed what we now call witness sets to represent positive dimensional solution sets of polynomial systems. Using monodromy and the linear trace these solution sets are decomposed into irreducible components, yielding a numerical irreducible decomposition.
In 1985, Erich Kaltofen presented a parallel NC algorithm for absolute factorization. Chanderjit Bajaj, John Canny, Thomas Garrity, and Joe Warren viewed the factorization of rational polynomials over the complex numbers as a topological problem (not an algebraic one) and presented in 1989 another parallel NC algorithm. Linear traces were first introduced in 1991 by T. Sasaki, M. Suzuki, M. Kolar, and M. Sasaki; and further developed by Andre Galligo and David Rupprecht, and most recently by Guillaume Cheze.
Joint with Anton Leykin, a first straightforward parallel implementation using the manager/worker paradigm suffered a decline in speedup for a growing number of processors because of the sequential overhead done on the manager node. A new monodromy breakup algorithm gives a second better performing parallel implementation.
Computational Complexity of Polynomial Factorization, American Institute of Mathematics, 15-19 May 2006.