关于欧几里得空间的Nash象
On Nash images of Euclidean spaces
摘要：
在这项工作中，我们的特点是纳什映射$f:{\mathbb R}^m\to{\mathbb R}^n$的图像的${\mathbb R}^n$的子集。证明了Shiota猜想，并证明了子集${\mathcal S}\subset{\mathbb R}^n$是Nash映射$f:{\mathbb R}^m\to{\mathbb R}^n$的像当且仅当${\mathcal S}$是维数$d\leq m$的半代数纯维数，且存在解析路$α:[0,1]\to{\mathcal S}$，其像满足${\mathcal S}$的正则点集的所有连通分支.一些显著的结果如下：（1）具有弧对称闭包的维数$d$的纯维不可约半代数集是PDMATHAMT的Nash象;（2）半代数集是不可约代数集的投影，其连通分支是Nash同构于欧几里得空间的;（3）紧的有边界的$d$ -维光滑流形是${\mathbb R}^d$的光滑象。
英文摘要：
In this work we characterize the subsets of ${\mathbb R}^n$ that are images of Nash maps $f:{\mathbb R}^m\to{\mathbb R}^n$. We prove Shiota's conjecture and show that a subset ${\mathcal S}\subset{\mathbb R}^n$ is the image of a Nash map $f:{\mathbb R}^m\to{\mathbb R}^n$ if and only if ${\mathcal S}$ is semialgebraic, pure dimensional of dimension $d\leq m$ and there exists an analytic path $α:[0,1]\to{\mathcal S}$ whose image meets all the connected components of the set of regular points of ${\mathcal S}$. Some remarkable consequences are the following: (1) pure dimensional irreducible semialgebraic sets of dimension $d$ with arc-symmetric closure are Nash images of ${\mathbb R}^d$; (2) semialgebraic sets are projections of irreducible algebraic sets whose connected components are Nash diffeomorphic to Euclidean spaces; and (3) compact $d$-dimensional smooth manifolds with boundary are smooth images of ${\mathbb R}^d$.
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