Let $S$ be a rational projective surface given by means of a projective rational parametrization whose base locus satisfies a mild assumption. In this paper we present an algorithm that provides three rational maps $f,g,h:\mathbb{A}^2 --\to S\subset \mathbb{P}^n$ such that the union of the three images covers $S$. As a consequence, we present a second algorithm that generates two rational maps $f,\tilde{g}:\mathbb{A}^2 --\to S$, such that the union of their images covers the affine surface $S\cap \mathbb{A}^n$. In the affine case, the number of rational maps involved in the cover is in general optimal.

中文翻译：

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用有理参数化图像覆盖有理曲面

设$ S $是通过投影有理参数化给出的有理投影面，其基本轨迹满足一个温和的假设。在本文中，我们提出了一种算法，该算法提供了三个有理映射$ f，g，h：\ mathbb {A} ^ 2-\到S \ subset \ mathbb {P} ^ n $，使得三个图像的并集覆盖$ S $。结果，我们提出了第二种算法，该算法生成两个有理映射$ f，\ tilde {g}：\ mathbb {A} ^ 2-\至S $，以使其图像的并集覆盖仿射表面$ S。 \ cap \ mathbb {A} ^ n $。在仿射情况下，封面涉及的有理图的数量通常是最佳的。