Methods for mechanical geometry formula deriving
ISSAC '90 Proceedings of the international symposium on Symbolic and algebraic computation
Gro¨bner bases: a computational approach to commutative algebra
Gro¨bner bases: a computational approach to commutative algebra
Algebraic and geometric reasoning using Dixon resultants
ISSAC '94 Proceedings of the international symposium on Symbolic and algebraic computation
Comparison of various multivariate resultant formulations
ISSAC '95 Proceedings of the 1995 international symposium on Symbolic and algebraic computation
Sparsity considerations in Dixon resultants
STOC '96 Proceedings of the twenty-eighth annual ACM symposium on Theory of computing
Extraneous factors in the Dixon resultant formulation
ISSAC '97 Proceedings of the 1997 international symposium on Symbolic and algebraic computation
Efficient variable elimination using resultants
Efficient variable elimination using resultants
An Efficient Algorithm for the Sparse Mixed Resultant
AAECC-10 Proceedings of the 10th International Symposium on Applied Algebra, Algebraic Algorithms and Error-Correcting Codes
On the efficiency and optimality of Dixon-based resultant methods
Proceedings of the 2002 international symposium on Symbolic and algebraic computation
Multihomogeneous resultant matrices
Proceedings of the 2002 international symposium on Symbolic and algebraic 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)
Multihomogeneous resultant formulae by means of complexes
Journal of Symbolic Computation - Special issue: International symposium on symbolic and algebraic computation (ISSAC 2002)
ISSAC '04 Proceedings of the 2004 international symposium on Symbolic and algebraic computation
Cayley-Dixon projection operator for multi-univariate composed polynomials
Journal of Symbolic Computation
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A structural criteria on polynomial systems is developed for which the generalized Dixon formulation of multivariate resultants defined by Kapur, Saxena and Yang (1994) computes the resultant exactly. The concept of a Dixon-exact support (the set of exponent vectors of terms appearing in a polynomial system) is introduced so that the Dixon formulation produces the exact resultant for generic unmixed polynomial systems whose support is Dixon-exact. A geometric operation, called direct-sum, on the supports is defined that preserves the property of supports being Dixon-exact. Generic n-degree systems and multigraded systems are shown to be a special case of generic unmixed polynomial systems whose support is Dixon-exact. Using a scaling techniques discussed by Kapur and Saxena (1997), a wide class of polynomial systems can be identified for which the Dixon formulation produces exact resultants. This analysis can be used to classify terms appearing in the convex hull (also called the Newton polytope) of the support of a polynomial system that can cause extraneous factors in the computation of a projection operation by the generalized Dixon formulation. For the bivariate case, a complete analysis of the terms corresponding to the exponent vectors in the Newton polytope of the support of a polynomial system is given vis a vis their role in producing extraneous factors in a projection operator. A necessary and sufficient condition is developed for a support to be Dixon-exact. Such an analysis is likely to give insights for the general case of elimination of arbitrarily many variables.