Polynomial Factorization Is Simple and Helpful -- More So Than It Seems to Be
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Univariate polynomial root-finding has been studied for four millennia, still stays important for modern computations, and remains the subject of intensive research. Hundreds if not thousands efficient polynomial root-finders have been proposed and analyzed. One of them, proposed in 1995, solves this problem within a nearly optimal time bound, that is, nearly as fast as one can access the input coefficients. It also solves (and also within a nearly optimal time bound) the related problem of approximate polynomial factorization, which has a number of other important applications besides root-finding. The algorithm, however, is much involved, not transparent, and hard to implement. By using its factorization techniques, however, we substantially improve the known efficient polynomial root-finders that rely on subdivision or on functional iterations.
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