Improved algorithms for computing determinants and resultants

AbstractOur 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

Macaulay style formulas for sparse resultants

We present formulas for computing the resultant of sparse polyno- mials as a quotient of two determinants, the denominator being a minor of the numerator. These formulas extend the original formulation given by Macaulay for homogeneous polynomials

Rational Univariate Reduction via toric resultants

AbstractWe describe algorithms for solving a given system of multivariate polynomial equations via the Rational Univariate Reduction (RUR). We compute the RUR from the toric resultant of the input system. Our algorithms derandomize several of the choices made in similar prior algorithms. We also propose a new derandomized algorithm for solving an overdetermined system. Finally, we analyze the computational complexity of the algorithm, and discuss its implementation and performance

A complete implementation for computing general dimensional convex hulls

Programme 2 - Calcul symbolique, programmation et genie logiciel - Projet SafirSIGLEAvailable at INIST (FR), Document Supply Service, under shelf-number : 14802 E, issue : a.1995 n.2551 / INIST-CNRS - Institut de l'Information Scientifique et TechniqueFRFranc

A Complete Implementation for Computing General Dimensional Convex Hulls

We study two important, and often complementary, issues in the implementation of geometric algorithms, namely exact arithmetic and degeneracy. We focus on integer arithmetic and propose a general and efficient method for its implementation based on modular arithmetic. We suggest that probabilistic modular arithmetic may be of wide interest, as it combines the advantages of modular arithmetic with randomization in order to speed up the lifting of residues to an integer. We derive general error bounds and discuss the implementation of this approach in our general-dimension convex hull program. The use of perturbations as a method to cope with input degeneracy is also illustrated. We present the implementation of a computationally efficient scheme that, moreover, greatly simplifies the task of programming. We concentrate on postprocessing, often perceived as the Achilles' heel of perturbations. Starting in the context of a specific application in robotics, we examine the complexity of p..