Efficient incremental algorithms for the sparse resultant and the mixed volume
Journal of Symbolic Computation
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
Multivariate polynomials, duality, and structured matrices
Journal of Complexity
Conditions for exact resultants using the Dixon formulation
ISSAC '00 Proceedings of the 2000 international symposium on Symbolic and algebraic computation
Symbolic and numeric methods for exploiting structure in constructing resultant matrices
Journal of Symbolic Computation
Topics in resultants and implicitization
Topics in resultants and implicitization
Multihomogeneous resultant formulae by means of complexes
Journal of Symbolic Computation - Special issue: International symposium on symbolic and algebraic computation (ISSAC 2002)
The resultant of an unmixed bivariate system
Journal of Symbolic Computation - Special issue: International symposium on symbolic and algebraic computation (ISSAC 2002)
On the complexity of computing gröbner bases for quasi-homogeneous systems
Proceedings of the 38th international symposium on International symposium on symbolic and algebraic computation
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Multihomogeneous structure in algebraic systems is the first step away from the classical theory of homogeneous equations towards fully exploiting arbitrary supports. We propose constructive methods for resultant matrices in the entire spectrum of resultant formulae, ranging from pure Sylvester to pure Bézout types, including hybrid matrices. Our approach makes heavy use of the combinatorics of multihomogeneous systems, inspired by and generalizing certain joint results by Zelevinsky, and Sturmfels or Weyman [15, 18]. One contribution is to provide conditions and algorithmic tools so as to classify and construct the smallest possible determinantal formulae for multihomogeneous resultants. We also examine the smallest Sylvester-type matrices, generically of full rank, which yield a multiple of the resultant. The last contribution is to characterize the systems that admit a purely Bézout-type matrix and show a bijection of such matrices with the permutations of the variable groups. Interestingly, it is the same class of systems admitting an optimal Sylvester-type formula. We conclude with an example showing all kinds of matrices that may be encountered, and illustrations of our MAPLE implementation.