On Nash images of Euclidean spaces

Abstract

In this work we characterize the subsets of Rn that are images of Nash maps f : Rm → Rn. We prove Shiota’s conjecture and show that a subset S ⊂ Rn is the image of a Nash map f : Rm → Rn if and only if S is semialgebraic, pure dimensional of dimension d ≤ m and there exists an analytic path α : [0, 1] → S whose image meets all the connected components of the set of regular points of S. Two remarkable consequences are the following: (1) pure dimensional irreducible semialgebraic sets of dimension d with arc-symmetric closure are Nash images of Rd; and (2) semialgebraic sets are projections of irreducible algebraic sets whose connected components are Nash diffeomorphic to Euclidean spaces.