On Nash images of Euclidean spaces

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 subsetS⊂Rnis the image of a Nash mapf:Rm→Rnif and only ifSis semialgebraic, pure dimensional of dimensiond≤mand there exists an analytic pathα:[0,1]→Swhose image meets all the connected components of the set of regular points ofS. 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.