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.)
Comparison of various multivariate resultant formulations
ISSAC '95 Proceedings of the 1995 international symposium on Symbolic and algebraic computation
Efficient incremental algorithms for the sparse resultant and the mixed volume
Journal of Symbolic Computation
Sparsity considerations in Dixon resultants
STOC '96 Proceedings of the twenty-eighth annual ACM symposium on Theory of computing
How good are convex hull algorithms?
Computational Geometry: Theory and Applications
A subdivision-based algorithm for the sparse resultant
Journal of the ACM (JACM)
Conditions for exact resultants using the Dixon formulation
ISSAC '00 Proceedings of the 2000 international symposium on Symbolic and algebraic computation
Rectangular corner cutting and Sylvester A-resultants
ISSAC '00 Proceedings of the 2000 international symposium on Symbolic and algebraic computation
Rectangular corner cutting and Dixon A -resultants
Journal of Symbolic Computation
Exact resultants for corner-cut unmixed multivariate polynomial systems using the Dixon formulation
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
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A geometric concept of the support hull of the support of a polynomial was used earlier by the authors for developing a tight upper bound on the size of the Cayley-Dixon resultant matrix for an unmixed polynomial system. The relationship between the support hull and the Cayley-Dixon resultant construction is analyzed in this paper. The support hull is shown to play an important role in the construction and analysis of resultant matrices based on the Cayley-Dixon formulation, similar to the role played by the associated convex hull (Newton polytope) for analyzing resultant matrices over the toric variety. For an unmixed polynomial system, the sizes of the resultant matrices (both dialytic as well as nondialytic) constructed using the Cayley-Dixon formulation are determined by the support hull of its support. Consequently, degree of the projection operator (which is in general, a nontrivial multiple of the resultant) computed from such a resultant matrix is determined by the support hull.The support hull of a given support is similar to its convex hull except that instead of the Euclidean distance, the support hull is defined using rectilinear distance. The concept of a support-hull interior point is introduced. It is proved that for an unmixed polynomial system, the size of the resultant matrix (both dialytic and nondialytic) based on the Cayley-Dixon formulation remains the same even if a term whose exponent is support-hull interior with respect to the support is generically added to the polynomial system. This key insight turned out to be instrumental in generalizing the concept of an unmixed polynomial system with a corner-cut support from 2 dimensions to arbitrary dimension as well as identifying an unmixed polynomial system with almost corner-cut support in arbitrary dimension.An algorithm for computing the size (and the lattice points) of the support hull of a given support is presented. It is proved that determining whether a given lattice point is not in the support hull, is NP-complete. A heuristic for computing a good variable ordering for constructing Dixon matrices for mixed as well as unmixed polynomial systems is proposed using the support hull and its projections. This is one of the first results on developing heuristics for variable orderings for constructing resultant matrices. A construction for a Sylvester-type resultant matrix based on the support hull of a polynomial system is also given.