STOC '95 Proceedings of the twenty-seventh annual ACM symposium on Theory of computing
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The author deduces some new probabilistic estimates on the distances between the zeros of a polynomial $p(x)$ by using some properties of the discriminant of $p(x)$ and applies these estimates to improve the fastest deterministic algorithm for approximating polynomial factorization over the complex field. Namely, given a natural $n$, positive $\epsilon$, such that $\log (1/\epsilon) = O(n\log n)$, and the complex coefficients of a polynomial $p(x)=\sum^n_{i=0} p_ix^i$, such that $p_n\ne 0$, $\sum _i |p_i| \le 1$, a factorization of $p(x)$ (within the error norm $\epsilon$) is computed as a product of factors of degrees at most $n/2$, by using $O(\log^2n)$ time and $n^3$ processors under the PRAM arithmetic model of parallel computing or by using $O(n^2\log^2 n)$ arithmetic operations. The algorithm is randomized, of Las Vegas type, allowing a failure with a probability at most $\delta$, for any positive $\delta