On Nash images of Euclidean spaces

Abstract

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 $\alpha:[0,1]\to{\mathcal S}$ 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 ${\mathbb R}^d$.