Properness and Inversion of Rational Parametrizations of Surfaces
Properness and Inversion of Rational Parametrizations of Surfaces
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DOI：
10.1007/s002000100089
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发表时间：
2002-04
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影响因子：
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通讯作者：
S. Pérez-Díaz;J. Schicho;J. Sendra
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文献类型：
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作者：
S. Pérez-Díaz;J. Schicho;J. Sendra
在本文中，我们通过有理函数域上的参数化直接生成的一些附加代数超曲面的交点的存在性来表征超曲面有理参数化的正确性。更准确地说，如果V 是代数闭域上的超曲面？特征零和 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}\end{document} 是 V 的有理参数化，则表征是根据由 xiqi(t´)−pi(t´),i=1,...,n 定义的超曲面的交点给出的，而不是 ?(V) 的代数闭包。此外，对于表面的情况，我们展示了如何通过算法来表达这些结果。因此，我们提出了一个算法标准来决定给定的理性参数化是否正确。此外，如果参数化正确，算法还会计算参数化的逆。此外，对于曲面，辅助超曲面变成了 ?(V) 上的平面曲线，因此该算法本质上是基于合力的。我们已经实现了这些想法，并且我们将我们的方法与基于 Gröbner 基的方法进行了实证比较。
In this paper we characterize the properness of rational parametrizations of hypersurfaces by means of the existence of intersection points of some additional algebraic hypersurfaces directly generated from the parametrization over a field of rational functions. More precisely, ifVis a hypersurface over an algebraically closed field ? of characteristic zero and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}\end{document} is a rational parametrization ofV, then the characterization is given in terms of the intersection points of the hypersurfaces defined byxiqi(t¯)−pi(t¯),i=1,...,nover the algebraic closure of ?(V). In addition, for the case of surfaces we show how these results can be stated algorithmically. As a consequence we present an algorithmic criteria to decide whether a given rational parametrization is proper. Furthermore, if the parametrization is proper, the algorithm also computes the inverse of the parametrization. Moreover, for surfaces the auxiliary hypersurfaces turn to be plane curves over ?(V), and hence the algorithm is essentially based on resultants. We have implemented these ideas, and we have empirically compared our method with the method based on Gröbner basis.