Resultants and Implicitization by Moving Surfaces

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The method of moving surfaces is a fundamentally new and significantly better method for implicitizing rational surfaces. This method has recently been applied to address the problem of multivariate polynomial resultants. Initial successes include finding determinental formulae for some sparse resultants. The method of moving surfaces produces expressions that are more compact than previous ones, and unlike previous determinental formulae do not contain extraneous factors. Multivariate resultants provide a powerful computational tool for solving problems involving polynomial equations in several variables. While Groebner bases can also be used to solve systems of polynomial equations (as well as answer additional questions about polynomial systems), multivariate resultants are generally more efficient at eliminating variables than Groebner bases. Preliminary examples suggest that the method of moving surfaces might be even more efficient than current multivariate resultant techniques. The method of moving surfaces can be used to attack a wide variety of problems in elimination theory including: 1. constructing multivariate resultants without extraneous factors 2. building sparse resultants 3. implicitizing rational curves and surfaces 4. developing inversion formulas for rational curves and surfaces 5. intersecting rational curves and surfaces 6. intersecting algebraic curves and surfaces. The method of moving curves has already lead to a new classification scheme for rational curves; a similar scheme for rational surfaces will be investigated. While rigorous theoretical foundation for the method of moving curves based on ideal theory has been developed, the method of moving surfaces is currently substantiated only by several Mathematica examples and a few preliminary theorems. Thus the theoretical foundation is far from complete. For example, the method provides an algorithm for determining the rows of a matrix whose determinant is a sparse resultant if it does not vanish. So far, the method has never failed to produce a non-vanishing determinant, but no proof exists that the method will never fail. A constructive proof will be sought. The method has thus far been applied to eliminating one variable from two polynomials, or two variables from three polynomials. The general problem of eliminating n-1 variables from n polynomial equations will be studied.