We use numerical homotopy continuation methods to obtain approximate solutions to systems of polynomial equations. A solution is called singular when the Jacobian matrix of the polynomial system is singular at the solution. Because Newton's method (as used in continuation methods) fails when the Jacobian matrix is singular, singular solutions are challenging to numerical solvers.
In this talk we will survey recent progress on dealing with singular solutions, as they occur as isolated roots of high multiplicity (joint work with Anton Leykin and Ailing Zhao) or as they occur as nonisolated roots belonging to positive dimensional solution sets (joint work with Andrew Sommese and Charles Wampler). In both instances, the solutions are "symbolic-numeric": they are not merely approximate numbers, but satisfy extra equations, which are part of the description of the solution sets.
As software is the practical foundation of computational mathematics, we will emphasize the recent updates in PHCpack, a software package to solve polynomial systems using homotopy continuation methods.
Real Number Complexity Workshop, FoCM'05, July 7-9, 2005, Universidad de Cantabria, Santander, Spain.