Resultants and Implicitization by Moving Surfaces

移动曲面法是一种全新的、明显更好的有理曲面隐式化方法。该方法最近被应用于求解多元多项式结果的问题。最初的成功包括找到一些稀疏结果的行列式公式。移动曲面的方法产生的表达式比以前的表达式更紧凑，并且不像以前的行列式公式包含外来因素。多元结果为解决涉及多变量多项式方程的问题提供了一个强大的计算工具。虽然格罗布纳基也可以用来求解多项式方程组（以及回答关于多项式系统的其他问题），但在消除变量方面，多元结果通常比格罗布纳基更有效。初步的例子表明，移动表面的方法可能比目前的多元合成技术更有效。移动曲面的方法可以用来解决消除理论中的各种各样的问题，包括：1。构造无外来因子的多元结果构建稀疏结果隐式有理曲线和曲面4。建立有理曲线曲面的反演公式有理曲线与曲面相交代数曲线和曲面的相交。移动曲线的方法已经产生了一种新的有理曲线分类方案；我们将研究有理曲面的类似格式。虽然基于理想理论的运动曲线的方法已经有了严格的理论基础，但是运动曲面的方法目前还只是通过几个Mathematica的例子和一些初步的定理来证实的。因此，理论基础还很不完善。例如，该方法提供了一种算法来确定矩阵的行，如果矩阵的行列式不消失，则矩阵的行列式是稀疏结果。到目前为止，这个方法从来没有产生一个不消失的行列式失败过，但是没有证据证明这个方法永远不会失败。将寻求建设性的证据。迄今为止，该方法已被应用于从两个多项式中消除一个变量，或从三个多项式中消除两个变量。本文将研究从n个多项式方程中消去n-1个变量的一般问题。

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.