Computer algebra: symbolic and algebraic computation (2nd ed.)
Computer algebra: symbolic and algebraic computation (2nd ed.)
Solving sparse linear equations over finite fields
IEEE Transactions on Information Theory
A deterministic algorithm for sparse multivariate polynomial interpolation
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
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
Estimating the largest eigenvalues by the power and Lanczos algorithms with a random start
SIAM Journal on Matrix Analysis and Applications
Probabilistic Bounds on the Extremal Eigenvalues and Condition Number by the Lanczos Algorithm
SIAM Journal on Matrix Analysis and Applications
On the Newton Polytope of the Resultant
Journal of Algebraic Combinatorics: An International Journal
Polynomial and matrix computations (vol. 1): fundamental algorithms
Polynomial and matrix computations (vol. 1): fundamental algorithms
Efficient incremental algorithms for the sparse resultant and the mixed volume
Journal of Symbolic Computation
On the complexity of sparse elimination
Journal of Complexity
Parallel computation of polynomial GCD and some related parallel computations over abstract fields
Theoretical Computer Science
The structure of sparse resultant matrices
ISSAC '97 Proceedings of the 1997 international symposium on Symbolic and algebraic computation
Solving special polynomial systems by using structured matrices and algebraic residues
FoCM '97 Selected papers of a conference on Foundations of computational mathematics
Asymptotic acceleration of solving multivariate polynomial systems of equations
STOC '98 Proceedings of the thirtieth annual ACM symposium on Theory of computing
The symmetric eigenvalue problem
The symmetric eigenvalue problem
The complexity of the matrix eigenproblem
STOC '99 Proceedings of the thirty-first annual ACM symposium on Theory of computing
LAPACK Users' guide (third ed.)
LAPACK Users' guide (third ed.)
A subdivision-based algorithm for the sparse resultant
Journal of the ACM (JACM)
Multivariate polynomials, duality, and structured matrices
Journal of Complexity
Computation of approximate polynomial GCDs and an extension
Information and Computation
Structured matrices and polynomials: unified superfast algorithms
Structured matrices and polynomials: unified superfast algorithms
Solving Systems of Polynomial Equations
IEEE Computer Graphics and Applications
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
Improved Sparse Multivariate Polynomial Interpolation Algorithms
ISAAC '88 Proceedings of the International Symposium ISSAC'88 on Symbolic and Algebraic Computation
An Algorithm for the Newton Resultant
An Algorithm for the Newton Resultant
New Techniques for the Computation of Linear Recurrence Coefficients
Finite Fields and Their Applications
Multihomogeneous resultant matrices
Proceedings of the 2002 international symposium on Symbolic and algebraic computation
Multihomogeneous resultant formulae by means of complexes
Journal of Symbolic Computation - Special issue: International symposium on symbolic and algebraic computation (ISSAC 2002)
Parametrization of approximate algebraic curves by lines
Theoretical Computer Science - Algebraic and numerical algorithm
Parametrization of approximate algebraic surfaces by lines
Computer Aided Geometric Design
Improved algorithms for computing determinants and resultants
Journal of Complexity - Special issue: Foundations of computational mathematics 2002 workshops
Multivariate power series multiplication
Proceedings of the 2005 international symposium on Symbolic and algebraic computation
Distance bounds of ε-points on hypersurfaces
Theoretical Computer Science
Solving over-determined systems by the subresultant method (with an appendix by Marc Chardin)
Journal of Symbolic Computation
Rational Univariate Reduction via toric resultants
Journal of Symbolic Computation
Schur aggregation for linear systems and determinants
Theoretical Computer Science
Parametrization of approximate algebraic surfaces by lines
Computer Aided Geometric Design
Algebraic and numerical algorithms
Algorithms and theory of computation handbook
Applications of FFT and structured matrices
Algorithms and theory of computation handbook
New progress in real and complex polynomial root-finding
Computers & Mathematics with Applications
Using a bihomogeneous resultant to find the singularities of rational space curves
Journal of Symbolic Computation
Implicitization of curves and (hyper)surfaces using predicted support
Theoretical Computer Science
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Resultant characterize the existence of roots of systems of multivariate nonlinear polynomial equations, while their matrices reduce the computation of all common zeros to a problem in linear algebra. Sparse elimination theory has introduced the sparse (or toric) resultant, which takes into account the sparse structure of the polynomials. The construction of sparse resultant, or Newton, matrices is the critical step in the computation of the multivariate resultant and the solution of a nonlinear system. We reveal and exploit the quasi-Toeplitz structure of the Newton matrix, thus decreasing the time complexity of constructing such matrices by roughly one order of magnitude to achieve quasi-quadratic complexity in the matrix dimension. The space complexity is also decreased analogously. These results imply similar improvements in the complexity of computing the resultant polynomial itself and of solving zero-dimensional systems. Our approach relies on fast vector-by-matrix multiplication and uses the following two methods as building blocks. First, a fast and numerically stable method for determining the rank of rectangular matrices, which works exclusively over floating point arithmetic. Second, exact polynomial arithmetic algorithms that improve upon the complexity of polynomial multiplication under our model of sparseness, offering bounds linear in the number of variables and the number of non-zero terms.