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What's the best way to find the roots of an arbitrary polynomial over the complex numbers?

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what do you mean by "find"? what sort of description do you want? numerical approximation, radical expression if it exists, minimal polynomial, ...?
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Use Newton's method starting from a close approximation.  

First make sure your polynomial has no repeated roots.  If you're working with polynomials f(x) with rational coefficients and gcd(f(x),f'(x)) is a nonzero constant then f(x) has no repeated roots. In general, if f(x) with rational coefficients has repeated roots then you can pass to a factor with the same roots all of multiplicity 1 by using the ratio f(x)/gcd(f(x),f'(x)).
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I don’t know if it is standard, but I know winding numbers can be used to find roots of arbitrary complex functions. If you have an initial bounding box, you can divide the remaining area to search in half at every stage of such an algorithm.

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