Computation of the degree of rational surface parametrizations
Computation of the degree of rational surface parametrizations
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DOI：
10.1016/j.jpaa.2004.02.011
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发表时间：
2004-10
影响因子：
0.8
通讯作者：
S. Pérez-Díaz;J. Sendra
中科院分区：
文献类型：
--
作者：
S. Pérez-Díaz;J. Sendra
代数曲面的有理仿射参数化建立仿射平面与曲面的有理对应。我们考虑这样一个有理映射的度的计算问题。一般来说，有理映射的度的确定可以通过消元法来实现。对于曲线，它表明，可以通过gcd计算计算的程度。在本文中，我们证明了由曲面参数化导出的有理映射的度可以通过广义CD和单变量结式计算来计算。其基本思想是将一般纤维的元素表示为直接从参数化构造的某些曲线的多个交点，并在有理函数域的代数闭包上定义。
A rational affine parametrization of an algebraic surface establishes a rational correspondence of the affine plane with the surface. We consider the problem of computing the degree of such a rational map. In general, determining the degree of a rational map can be achieved by means of elimination theoretic methods. For curves, it is shown that the degree can be computed by gcd computations. In this paper, we show that the degree of a rational map induced by a surface parametrization can be computed by means of gcd and univariate resultant computations. The basic idea is to express the elements of a generic fibre as the finitely many intersection points of certain curves directly constructed from the parametrization, and defined over the algebraic closure of a field of rational functions.