Computer algebra: symbolic and algebraic computation (2nd ed.)
Solving sparse linear equations over finite fields
IEEE Transactions on Information Theory
Some algebraic and geometric computations in PSPACE
STOC '88 Proceedings of the twentieth annual ACM symposium on Theory of computing
Solving systems of nonlinear polynomial equations faster
ISSAC '89 Proceedings of the ACM-SIGSAM 1989 international symposium on Symbolic and algebraic computation
Generalised characteristic polynomials
Journal of Symbolic Computation - Special issue on computational algebraic complexity
Processor efficient parallel solution of linear systems over an abstract field
SPAA '91 Proceedings of the third annual ACM symposium on Parallel algorithms and architectures
Implicit representation of rational parametric surfaces
Journal of Symbolic Computation
Multipolynomial resultant algorithms
Journal of Symbolic Computation
A bibliography on roots of polynomials
Journal of Computational and Applied Mathematics
Solving homogeneous linear equations over GF(2) via block Wiedemann algorithm
Mathematics of Computation
The theory of Smale's point estimation and its applications
Proceedings of the international meeting on Linear/nonlinear iterative methods and verification of solution
Efficient incremental algorithms for the sparse resultant and the mixed volume
Journal of Symbolic Computation
Matrix computations (3rd ed.)
Sign determination in residue number systems
Theoretical Computer Science - Special issue on real numbers and computers
Computing the isolated roots by matrix methods
Journal of Symbolic Computation - Special issue on symbolic numeric algebra for polynomials
Modern computer algebra
Fast deterministic computation of determinants of dense matrices
ISSAC '99 Proceedings of the 1999 international symposium on Symbolic and algebraic computation
A subdivision-based algorithm for the sparse resultant
Journal of the ACM (JACM)
Polynomial root finding using iterated Eigenvalue computation
Proceedings of the 2001 international symposium on Symbolic and algebraic computation
Structured matrices and polynomials: unified superfast algorithms
Structured matrices and polynomials: unified superfast algorithms
Symbolic and numeric methods for exploiting structure in constructing resultant matrices
Journal of Symbolic Computation
Point estimation of simultaneous methods for solving polynomial equations: a survey
Journal of Computational and Applied Mathematics
The Design and Analysis of Computer Algorithms
The Design and Analysis of Computer Algorithms
Univariate polynomials: nearly optimal algorithms for numerical factorization and root-finding
Journal of Symbolic Computation - Computer algebra: Selected papers from ISSAC 2001
On Wiedemann's Method of Solving Sparse Linear Systems
AAECC-9 Proceedings of the 9th International Symposium, on Applied Algebra, Algebraic Algorithms and Error-Correcting Codes
Randomized Acceleration of Fundamental Matrix Computations
STACS '02 Proceedings of the 19th Annual Symposium on Theoretical Aspects of Computer Science
An output-sensitive variant of the baby steps/giant steps determinant algorithm
Proceedings of the 2002 international symposium on Symbolic and algebraic computation
On computing the determinant and Smith form of an integer matrix
FOCS '00 Proceedings of the 41st Annual Symposium on Foundations of Computer Science
On square-free decomposition algorithms
SYMSAC '76 Proceedings of the third ACM symposium on Symbolic and algebraic computation
A method computing multiple roots of inexact polynomials
ISSAC '03 Proceedings of the 2003 international symposium on Symbolic and algebraic computation
High-order lifting and integrality certification
Journal of Symbolic Computation - Special issue: International symposium on symbolic and algebraic computation (ISSAC 2002)
The shifted number system for fast linear algebra on integer matrices
Journal of Complexity - Festschrift for the 70th birthday of Arnold Schönhage
Bernstein-Bezoutian matrices and curve implicitization
Theoretical Computer Science
Faster inversion and other black box matrix computations using efficient block projections
Proceedings of the 2007 international symposium on Symbolic and algebraic computation
Additive preconditioning and aggregation in matrix computations
Computers & Mathematics with Applications
Schur aggregation for linear systems and determinants
Theoretical Computer Science
The shifted number system for fast linear algebra on integer matrices
Journal of Complexity - Festschrift for the 70th birthday of Arnold Schönhage
Algebraic and numerical algorithms
Algorithms and theory of computation handbook
Proceedings of the 36th international symposium on Symbolic and algebraic computation
Univariate real root isolation in multiple extension fields
Proceedings of the 37th International Symposium on Symbolic and Algebraic Computation
Proceedings of the 38th international symposium on International symposium on symbolic and algebraic computation
Sub-linear root detection, and new hardness results, for sparse polynomials over finite fields
Proceedings of the 38th international symposium on International symposium on symbolic and algebraic computation
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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 Sylvestertype 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.