Surjective Nash maps between semialgebraic sets

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)}.