Homotopies and polynomial system solving I: basic principles

We present a survey of some basic ideas involved in the use of homotopies for solving systems of polynomial equations. These ideas are illustrated with many concrete examples. An introductory section on systems of polynomial equations and their solutions contains some necessary terminology that will be used in the sequel. We also describe a general algorithm to solve polynomial systems of n equations in n unknowns using homotopies. A Maple V implementation of the algorithm as well as a few accompanying Maple 6 worksheets are publicly available from the author's web page.