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On fast multiplication of polynomials over arbitrary algebras. (English) Zbl 0766.68055
We generalize the well-known Schönhage-Strassen algorithm [A. Schönhage and V. Strassen, Computing 7, 281–292 (1972; Zbl 0223.68007)] for multiplying large integers to an algorithm for multiplying polynomials with coefficients from an arbitrary, not necessarily commutative, not necessarily associative, algebra \({\mathcal A}\). Our main result is an algorithm to multiply polynomials of degree \(<n\) in \(O(n\log n)\) algebra multiplications and \(O(n\log n\log\log n)\) algebra additions/subtractions (we count a subtraction as an addition). The constant implied by the “\(O\)” does not depend upon the algebra \({\mathcal A}\). The parallel complexity of our algorithm, i.e., the depth of the corresponding arithmetic circuit, is \(O(\log n)\).
Reviewer: D.G.Cantor

MSC:
68W30 Symbolic computation and algebraic computation
68W10 Parallel algorithms in computer science
13P05 Polynomials, factorization in commutative rings
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