Improved algorithms for computing determinants and resultants

Our first contribution is a substantial acceleration of randomized computation of scalar, univariate, and multivariate matrix determinants, in terms of the output-sensitive bit operation complexity bounds, including computation modulo a product of random primes from a fixed range. This acceleration is dramatic in a critical application, namely solving polynomial systems and related studies, via computing the resultant. This is achieved by combining our techniques with the primitive-element method, which leads to an effective implicit representation of the roots. We systematically examine quotient formulae of Sylvester-type resultant matrices, including matrix polynomials and the u-resultant. We reduce the known bit operation complexity bounds by almost an order of magnitude, in terms of the resultant matrix dimension. Our theoretical and practical improvements cover the highly important cases of sparse and degenerate systems.