New resultant inequalities and complex polynomial factorization

Abstract

We deduce 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 apply these estimates to improve the fastest deterministic algorithm for approximating polynomial factorization over the complex field. Namely, given a natural n, positive ε, such that log(1/ε)=O(n log n), and the complex coefficients of a polynomial p(x) = Σ ⁿ_i=0 p _ixⁱ, such that P _n ≠ 0, ∑_i¦p _i¦≤1, we compute (within the error norm ε) a factorization of p(x) as a product of factors of degrees at most n/2, by using O(log² n) time and n ³ processors under the PRAM arithmetic model of parallel computing or by using O(n ² log² n) arithmetic operations. The algorithm is randomized, of Las Vegas type, allowing a failure with a probability at most δ, for any positive δ < 1 such that log(1/δ)=O(log n). Except for a narrow class of polynomials p(x), these results can be also obtained for ε such that log(1/ε)=O(n ² log n).

Supported by NSF Grant CCR 9020690 and PSC-CUNY Award 662478

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