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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.

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    $\begingroup$ This question is a little subtle because the property of having isolated real roots is not stable. Perturb it one way and the root disappears. Perturb it the other way and the root expands into a sphere of roots. That, said, at an isolated real zero, all the partial derivatives vanish. Thus I would search for locations where all partial derivatives vanish. So I think the question is very much related to finding common zeroes of $n$ polynomials in $n$ variables - which you can actually use Newton's method for! A similar way to find all the common zeroes is not obvious, though. $\endgroup$ – Will Sawin Aug 11 '13 at 3:36
  • $\begingroup$ You could apply gloptipoly to the squared polynomial. $\endgroup$ – David Ketcheson Oct 7 '13 at 9:56
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Gröbner basis methods will help you find all roots. There is no efficient method, though; all methods have really bad worst-case complexity.

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