If I am asked to find the roots of a polynomial of one variable, I will use a computer to estimate the eigenvalues of its companion matrix. Now suppose I'm given a real polynomial of multiple variables, and I'm promised that the roots form a discrete set. Is there a similarly fast and stable way to find the roots? Considering Denis Serre's answer to this question, I doubt Newton's method will be as helpful.
Gröbner basis methods will help you find all roots. There is no efficient method, though; all methods have really bad worst-case complexity.