In engineering and applied mathematics, polynomial systems arise whose solution sets contain components of different dimensions and multiplicities. We present algorithms, based on homotopy continuation, that compute much of the geometric information contained in the primary decomposition of the solution set. In particular, ignoring multiplicities, our algorithms lay out the decomposition of the set of solutions into irreducible components, by finding, at each dimension, generic points on each component. As by-products, the computation also determines the degree of each component and an upper bound on its multiplicity. The bound is sharp (i.e., equal to one) for reduced components. The algorithms make essential use of generic projection and interpolation, and can, if desired, describe each irreducible component precisely as the common zeroes of a finite number of polynomials.
This is a joint work with Andrew J. Sommese (University of Notre Dame) and Charles W. Wampler (General Motors Research Laboratories).
Workshop on Symbolic Computation: Solving Equations in Algebra, Geometry and Engineering, June 11-15, Mt Holyoke College.