关于其像是双曲线与实仿射空间乘积所围成区域的实代数映射
On real algebraic maps whose images are domains surrounded by the products of hyperbolas and real affine spaces
AI总结：
本文显式构造了其像由双曲线连通分支与实仿射空间乘积所围成区域的实代数映射，并讨论了从递增实数序列获得实多项式零点集的相关问题，推动了Reeb图实现与实代数几何中的显式构造研究。
摘要：
此前，我们已经系统地构造了显式的实代数函数，这些函数表示为光滑实代数映射的复合，其像是被次数为1或2的超曲面与典范投影所围成的区域。在此，我们给出新的例子，其中的每个超曲面显式地是一条双曲线的一个连通分支与一维仿射空间的一份拷贝的乘积。作为相关的未来工作，我们还讨论了从递增实数序列显式地获得某些实多项式零点集的问题。这项工作受到Sharko首先提出的光滑函数理论中一个问题的推动：我们能否构造出好的光滑函数，使得其Reeb图如预先规定的那样？光滑函数的Reeb空间是自然得到的图，其底空间是流形由原像的连通分支构成的商空间。作者首先考虑了尊重原像拓扑的变体，并在之前得到了若干结果。我们的工作也受到由Nash开创的实代数几何的推动。我们可以知道光滑流形及某些一般集合的实代数结构的存在性，并且我们已经知道光滑映射被实代数映射逼近的若干结果。我们的兴趣在于显式构造，而这是困难的。
英文摘要：
Previously, we have systematically constructed explicit real algebraic functions which are represented as the compositions of smooth real algebraic maps whose images are domains surrounded by hypersurfaces of degree 1 or 2 with canonical projections. Here we give new examples with the hypersurfaces each of which is the product of a connected component of a hyperbola and a copy of the $1$-dimensional affine space explicitly. As a related future work we also discuss problems to obtain the zero sets of some real polynomials explicitly from increasing sequences of real numbers. This is motivated by a problem in theory of smooth functions proposed first by Sharko: can we construct nice smooth functions whose Reeb graphs are as prescribed? The Reeb space of a smooth function is the naturally obtained graph whose underlying space is the quotient space of the manifold consisting of connected components of preimages. The author first considered variants respecting the topologies of the preimages and obtained several results before. Our work is also motivated by real algebraic geometry, pioneered by Nash. We can know existence of real algebraic structures of smooth manifolds and some general sets and we already know several approximations of smooth maps by real algebraic maps. Our interest lies in explicit construction, which is difficult.
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