Generalised characteristic polynomials
Journal of Symbolic Computation - Special issue on computational algebraic complexity
Algebraic and geometric reasoning using Dixon resultants
ISSAC '94 Proceedings of the 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
Algebraic pruning: a fast technique for curve and surface intersection
Computer Aided Geometric Design
Generalized resultants over unirational algebraic varieties
Journal of Symbolic Computation - Special issue on symbolic computation in algebra, analysis and geometry
ACM Transactions on Mathematical Software (TOMS)
Polynomials and Linear Control Systems
Polynomials and Linear Control Systems
Computational experiments with resultants for scaled Bernstein polynomials
Mathematical Methods for Curves and Surfaces
Journal of Symbolic Computation
MANIPULATING POLYNOMIALS IN GENERALIZED FORM
MANIPULATING POLYNOMIALS IN GENERALIZED FORM
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
Theoretical Computer Science - Algebraic and numerical algorithm
On computing polynomial GCDs in alternate bases
Proceedings of the 2006 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
Comparing acceleration techniques for the Dixon and Macaulay resultants
Mathematics and Computers in Simulation
Cayley-Dixon resultant matrices of multi-univariate composed polynomials
CASC'05 Proceedings of the 8th international conference on Computer Algebra in Scientific Computing
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Macaulay and Dixon resultant formulations are proposed for parametrized multivariate polynomial systems represented in Bernstein basis. It is proved that the Macaulay resultant for a polynomial system in Bernstein basis vanishes for the total degree case if and only if the either the polynomial system has a common Bernstein-toric root, a common infinite root, or the leading forms of the polynomial system obtained by replacing every variable xi in the original polynomial system by yi/1+yi have a non-trivial common root. For the Dixon resultant formulation, the rank sub-matrix constructions for the original system and the transformed system are shown to be essentially equivalent. Known results about exactness of Dixon resultants of a sub-class of polynomial systems as discussed in Chtcherba and Kapur in Journal of Symbolic Computation (August, 2003) carry over to polynomial systems represented in the Bernstein basis. Furthermore, in certain cases, when the extraneous factor in a projection operator constructed from the Dixon resultant formulation is precisely known, such results also carry over to projection operators of polynomial systems in the Bernstein basis where extraneous factors are precisely known. Applications of these results in the context of geometry theorem proving, implicitization and intersection of surfaces with curves are discussed. While Macaulay matrices become large when polynomials in Bernstein bases are used for problems in these applications, Dixon matrices are roughly of the same size.