On Nash images of Euclidean spaces

TL;DR

This paper characterizes subsets of Euclidean spaces that are Nash images, proving Shiota's conjecture and establishing conditions involving semialgebraic sets, analytic paths, and regular points.

Contribution

It proves Shiota's conjecture, providing a complete characterization of Nash images of Euclidean spaces and deriving several significant consequences.

Findings

Characterization of Nash images via semialgebraic, pure dimensional sets and analytic paths.

Proof of Shiota's conjecture confirming the characterization.

Identification of conditions under which semialgebraic sets are Nash images.

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 ${\mathbb R}^d$.