Algebraic and numeric techniques in modeling and robotics
Algebraic and numeric techniques in modeling and robotics
On the Newton Polytope of the Resultant
Journal of Algebraic Combinatorics: An International Journal
On the complexity of sparse elimination
Journal of Complexity
Sparsity considerations in Dixon resultants
STOC '96 Proceedings of the twenty-eighth annual ACM symposium on Theory of computing
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)
Rectangular corner cutting and Sylvester A-resultants
ISSAC '00 Proceedings of the 2000 international symposium on Symbolic and algebraic computation
An Algorithm for the Newton Resultant
An Algorithm for the Newton Resultant
Hybrid sparse resultant matrices for bivariate polynomials
Journal of Symbolic Computation - Computer algebra: Selected papers from ISSAC 2001
Determinantal formula for the chow form of a toric surface
Proceedings of the 2002 international symposium on Symbolic and algebraic computation
Multihomogeneous resultant formulae for systems with scaled support
Proceedings of the 2009 international symposium on Symbolic and algebraic computation
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Our main contribution is an explicit construction of square resultant matrices, which are submatrices of those introduced by Cattani, Dickenstein and Sturmfels [4]. The determinant is a nontrivial multiple of the sparse (or toric) resultant. The matrix is hybrid in that it contains a submatrix of Sylvester type and an additional row expressing the toric Jacobian. If we restrict attention to such matrices, the algorithm yields the smallest possible matrix in general. This is achieved by strongly exploiting the combinatorics of sparse elimination. The algorithm uses a new piecewise-linear lifting, defined for bivariate systems of 3 polynomials with Newton polygons being scaled copies of a single polygon. The major motivation comes from systems encountered in CAD. Our MAPLE implementation, applied to certain examples, illustrates our construction and compares with alternative matrices.