Investigation of the Dixon Resultants

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Dixon's formulation for simultaneously eliminating several variables from a system of multivariate polynomial equations will be investigated. Solving polynomials equations and identifying conditions on parameters under which the equations have common solutions are fundamental problems in symbolic computations which arise almost everywhere. Polynomials are used to model many phenomena - particularly in robotics, kinematics, computer vision, solid modeling, graphics, chemical equilibrium and other engineering, CAD-CAM design, etc. Dixon's formulation directly generalizes Bezout-Cayley's formulation of a resultant for eliminating a single variable. Dixon resultants have not been extensively investigated apparently because they presumably got overshadowed by Macaulay's formulation of resultants for simultaneously eliminating several variables. Implementations of Dixon resultants, Macaulay resultants and sparse resultants have revealed that Dixon resultants are more efficient to compute in practice. Dixon resultants will be analyzed with a particular emphasis on studying the relationship of the size and structure of the Dixon matrix to the input polynomial system. ***