Fast Parallel Computation of the Polynomial Remainder Sequence via Bézout and Hankel Matrices

Abstract

If $u(x)$ and $v(x)$ are polynomials of degree n and m, respectively, $m < n$, all the coefficients of the polynomials generated by the Euclidean scheme applied to $u(x)$ and $v(x)$ can be computed by using $O(\log^{3} n)$ parallel arithmetic steps and $n^{2}/ \log n$ processors over any field of characteristic 0 supporting FFT (Fast Fourier Transform). If the field does not support FFT the number of processors is increased by a factor of $\log \log n$; if the field does not allow division by $n!$ the number of processors is increased by a factor of n. This result is obtained by reducing the Euclidean scheme to computing the block triangular factorization of the Bezout matrix associated with $u(x)$ and $v(x)$. This approach is also extended to the evaluation of polynomial gcd (greatest common divisor) over any field of constants in $O(\log^{2} n)$ steps with the same number of processors.