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
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 implicitization algorithm for rational surfaces with no base points
Journal of Symbolic Computation
Hybrid sparse resultant matrices for bivariate systems
Proceedings of the 2001 international symposium on Symbolic and algebraic computation
An Algorithm for the Newton Resultant
An Algorithm for the Newton Resultant
The resultant of an unmixed bivariate system
Journal of Symbolic Computation - Special issue: International symposium on symbolic and algebraic computation (ISSAC 2002)
Inversion of parameterized hypersurfaces by means of subresultants
ISSAC '04 Proceedings of the 2004 international symposium on Symbolic and algebraic computation
Conditions for determinantal formula for resultant of a polynomial system
Proceedings of the 2006 international symposium on Symbolic and algebraic computation
On the geometry of parametrized bicubic surfaces
Journal of Symbolic Computation
Solving over-determined systems by the subresultant method (with an appendix by Marc Chardin)
Journal of Symbolic Computation
Multihomogeneous resultant formulae for systems with scaled support
Journal of Symbolic Computation
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We study systems of three bivariate polynomials whose Newton polygons are scaled copies of a single polygon. Our main contribution is to construct square resultant matrices, which are submatrices of those introduced by Cattani et al. (1998), and whose determinants are nontrivial multiples 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 our attention to matrices of (almost) Sylvester-type and systems as specified above, then the algorithm yields the smallest possible matrix in general. This is achieved by strongly exploiting the combinatorics of sparse elimination, namely by a new piecewise-linear lifting. The major motivation comes from systems encountered in geometric modeling. Our preliminary MAPLE implementation, applied to certain examples, illustrates our construction and compares it with alternative matrices.