Implicit representation of rational parametric surfaces
Journal of Symbolic Computation
Algebraic and geometric reasoning using Dixon resultants
ISSAC '94 Proceedings of the international symposium on Symbolic and algebraic computation
Basic algebraic geometry 1 (2nd, revised and expanded ed.)
Basic algebraic geometry 1 (2nd, revised and expanded ed.)
Sparse elimination and applications in kinematics
Sparse elimination and applications in kinematics
Efficient incremental algorithms for the sparse resultant and the mixed volume
Journal of Symbolic Computation
Testing stability by quantifier elimination
Journal of Symbolic Computation - Special issue: applications of quantifier elimination
Efficient variable elimination using resultants
Efficient variable elimination using resultants
Matrices in elimination theory
Journal of Symbolic Computation - Special issue on polynomial elimination—algorithms and applications
A subdivision-based algorithm for the sparse resultant
Journal of the ACM (JACM)
Generalized resultants over unirational algebraic varieties
Journal of Symbolic Computation - Special issue on symbolic computation in algebra, analysis and geometry
An implicitization algorithm for rational surfaces with no base points
Journal of Symbolic Computation
Hybrid sparse resultant matrices for bivariate polynomials
Journal of Symbolic Computation - Computer algebra: Selected papers from ISSAC 2001
The resultant of an unmixed bivariate system
Journal of Symbolic Computation - Special issue: International symposium on symbolic and algebraic computation (ISSAC 2002)
A new sylvester-type resultant method based on the dixon-bezout formulation
A new sylvester-type resultant method based on the dixon-bezout formulation
Multivariate resultants in Bernstein basis
ADG'08 Proceedings of the 7th international conference on Automated deduction in geometry
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Matrices constructed from a parameterized multivariate polynomial system are analyzed to ensure that such a matrix contains a condition for the polynomial system to have common solutions irrespective of whether its parameters are specialized or not. Such matrices include resultant matrices constructed using well-known methods for computing resultants over projective, toric and affine varieties. Conditions on these matrices are identified under which the determinant of a maximal minor of such a matrix is a nontrivial multiple of the resultant over a given variety. This condition on matrices allows a generalization of a linear algebra construction, called rank submatrix, for extracting resultants from singular resultant matrices, as proposed by Kapur, Saxena and Yang in ISSAC'94. This construction has been found crucial for computing resultants of non-generic, specialized multivariate polynomial systems that arise in practical applications. The new condition makes the rank submatrix construction based on maximal minor more widely applicable by not requiring that the singular resultant matrix have a column independent of the remaining columns. Unlike perturbation methods, which require introducing a new variable, rank submatrix construction is faster and effective. Properties and conditions on symbolic matrices constructed from a polynomial system are discussed so that the resultant can be computed as a factor of the determinant of a maximal non-singular submatrix.