Bivariate polynomial multiplication patterns

  • Arnold Schönhage
Invited Contributions
Part of the Lecture Notes in Computer Science book series (LNCS, volume 948)


Motivated by multiplication of numerical univariate polynomials with various kinds of truncation we study corresponding bivariate problems A(x, y)·B(x, y) = C(x, y) in the algebraic setting with indeterminate coefficients over suitable ground fields, counting essential multiplications only. The rectangular case concerning factors A, B with entries x i y j for i ≤ n, j≤ m, e. g. with m = n, has complexity (2n + 1)2. Here multiplication with single truncation, computing the product C(x,y) mod xn+1, or mod y n+1 , seems still to have the same full multiplication complexity, as is well-known for the univariate case, while the double truncation case mod (x n+1 , y n+1 ) admits the reduced upper bound 3n2 + 4n + 1, opposed to a lower bound of 2n2 + 4n + 1. We have a similar saving factor for the triangular case with factors A, B of total degree n to be multiplied mod (x n+1 ,x n y,...,y xn+1 ). There remains the issue to find the exact complexities of these multiplication problems.


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Copyright information

© Springer-Verlag Berlin Heidelberg 1995

Authors and Affiliations

  • Arnold Schönhage
    • 1
  1. 1.Institut für Informatik IIUniversität BonnBonnGermany

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