Resultants and Implicitization by Moving Surfaces

移动表面的结果和隐式化

• 批准号: 9712407
• 负责人: Thomas Sederberg
• 金额: $ 9.87万
• 依托单位: Brigham Young University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1997
• 资助国家: 美国
• 起止时间: 1997-09-15 至 2000-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9712407&HistoricalAwards=false
• 关键词: Resultants; Implicitization; Moving; Surfaces

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.

移动曲面方法是一种全新的、明显更好的隐含有理曲面的方法。这种方法最近被应用于解决多元多项式结式的问题。最初的成功包括找到了一些稀疏结式的行列式。移动曲面的方法产生的表达式比以前的更紧凑，并且与以前的行列式不同的是，它不包含外部因素。多元结式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Groebner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)，但多元结式在消除变量方面通常比Groebner基更有效。初步的例子表明，移动曲面的方法可能比目前的多元合成技术更有效。移动曲面法可以用来解决消去论中的各种问题，包括：1.构造无外在因素的多元结式2.构造稀疏结式3.隐含有理曲线曲面4.建立有理曲线曲面的反演公式5.有理曲线曲面相交6.代数曲线曲面相交.移动曲线的方法已经导致了有理曲线的一种新的分类方案；类似的方案将被研究用于有理曲面。虽然基于理想理论的移动曲线法已经发展了严密的理论基础，但移动曲面法目前只有几个数学例子和几个初步定理来证明。因此，理论基础还远远不完整。例如，该方法提供了一种算法，用于确定行列式是稀疏结式的矩阵的行，如果它不消失的话。到目前为止，该方法从未失败过产生一个不消失的行列式，但没有证据表明该方法永远不会失败。我们将寻求建设性的证据。到目前为止，该方法已应用于从两个多项式中消去一个变量，或从三个多项式中消去两个变量。研究从n个多项式方程中消去n-1个变量的一般问题。