The authors give a dictionary for translating the key concepts of algebraic geometry to define an irreducible decomposition into data structures used by numerical algorithms.
Homotopy continuation methods are efficient to approximate all isolated solutions of polynomial systems. This involves the use of computational geometry techniques to compute a homotopy with start points combined with numerical methods to follow the solution paths defined by the homotopies.
The authors show how can be used this capability as a blackbox device to solve systems which have positive-dimensional components of solutions and they describe a simple Maple procedure to call the blackbox solver of PHCpack. Via sampling and projecting they obtain a numerical elimination procedure and they illustrate this sampling on three-dimensional spatial mechanical linkage, using Maple as a plotting tool. They also develop a low level interface to call the Ada routines from C programs.
Finally, they list some of their major benchmark applications.
For the entire collection see [Zbl 1008.00013].