We study smooth rational closed embeddings of the real affine line into the real affine plane, that is algebraic rational maps from the real affine line to the real affine plane which induce smooth closed embeddings of the real euclidean line into the real euclidean plane. We consider these up to equivalence under the group of birational automorphisms of the real affine plane which are diffeomorphisms of its real locus. We show that in contrast with the situation in the categories of smooth manifolds with smooth maps and of real algebraic varieties with regular maps where there is only one equivalence class up to isomorphism, there are non-equivalent smooth rational closed embeddings up to such birational diffeomorphisms.

中文翻译：

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实仿射平面中直线的代数模型

我们研究了实仿射线到实仿射平面的平滑有理闭嵌入，即从实仿射线到实仿射平面的代数有理映射，将实欧几里得线平滑封闭嵌入到实欧几里得平面中。我们在实仿射平面的双有理自同构群下将这些视为等价，它们是其实轨迹的微分同胚。我们表明，与具有光滑映射的光滑流形和具有正则映射的实代数变体类别中的情况相比，其中只有一个等价类直到同构，有不等价的光滑有理闭嵌入直到这种双有理微分同构.