Title: Surjective Nash maps between semialgebraic sets Show Chinese title

Title: 从半代数集到半代数集的满射纳什映射

Abstract: In this work we study the existence of surjective Nash maps between two given semialgebraic sets ${\mathcal S}$ and ${\mathcal T}$. Some key ingredients are: the irreducible components ${\mathcal S}_i^*$ of ${\mathcal S}$ (and their intersections), the analytic-path connected components ${\mathcal T}_j$ of ${\mathcal T}$ (and their intersections) and the relations between dimensions of the semialgebraic sets ${\mathcal S}_i^*$ and ${\mathcal T}_j$. A first step to approach the previous problem is the former characterization done by the second author of the images of affine spaces under Nash maps. The core result of this article to obtain a criterion to decide about the existence of surjective Nash maps between two semialgebraic sets is: {\em a full characterization of the semialgebraic subsets ${\mathcal S}\subset{\mathbb R}^n$ that are the image of the closed unit ball $\overline{\mathcal B}_m$ of ${\mathbb R}^m$ centered at the origin under a Nash map $f:{\mathbb R}^m\to{\mathbb R}^n$}. The necessary and sufficient conditions that must satisfy such a semialgebraic set ${\mathcal S}$ are: {\em it is compact, connected by analytic paths and has dimension $d\leq m$}. Two remarkable consequences of the latter result are the following: (1) {\em pure dimensional compact irreducible arc-symmetric semialgebraic sets of dimension $d$ are Nash images of $\overline{\mathcal B}_d$}, and (2) {\em compact semialgebraic sets of dimension $d$ are projections of non-singular algebraic sets of dimension $d$ whose connected components are Nash diffeomorphic to spheres (maybe of different dimensions)}. Show Chinese abstract

Abstract: 在本工作中，我们研究两个给定的半代数集${\mathcal S}$和${\mathcal T}$之间的满射纳什映射的存在性。 一些关键要素是：${\mathcal S}_i^*$是${\mathcal S}$的不可约分支（以及它们的交集），解析路径连通分支${\mathcal T}_j$是${\mathcal T}$的（以及它们的交集）以及半代数集${\mathcal S}_i^*$和${\mathcal T}_j$的维数之间的关系。 解决前面问题的第一步是第二作者对仿射空间在Nash映射下的像的先前特征描述。 本文的核心结果是获得一个准则，用于判断两个半代数集之间是否存在满射的Nash映射：{\em 对半代数子集${\mathcal S}\subset{\mathbb R}^n$的完整描述，这些子集是原点中心的闭单位球$\overline{\mathcal B}_m$在 Nash 映射$f:{\mathbb R}^m\to{\mathbb R}^n$下的像${\mathbb R}^m$}。 满足该半代数集${\mathcal S}$的必要充分条件是：{\em 它是紧致的，由解析路径连接且具有维度$d\leq m$}。 后者结果的两个显著结论如下：(1){\em 维数为$d$的纯维数紧致不可约弧对称半代数集是$\overline{\mathcal B}_d$的Nash像}，和 (2){\em 维数为$d$的紧致半代数集是维数为$d$的非奇异代数集的投影，其连通分支与球面（可能维数不同）Nash同胚}。