On regulous and regular images of Euclidean spaces

Abstract: In this work we compare the semialgebraic subsets that are images of regulous maps with those that are images of regular maps. Recall that a map f : R n $\rightarrow$ R m is regulous if it is a rational map that admits a continuous extension to R n. In case the set of (real) poles of f is empty we say that it is regular map. We prove that if S $\subset$ R m is the image of a regulous map f : R n $\rightarrow$ R m , there exists a dense semialgebraic subset T $\subset$ S and a regular map g : R n $\rightarrow$ R m such that g(R n) = T. In case dim(S) = n, we may assume that the difference S \ T has codimension $\ge$ 2 in S. If we restrict our scope to regulous maps from the plane the result is neat: if f : R 2 $\rightarrow$ R m is a regulous map, there exists a regular map g : R 2 $\rightarrow$ R m such that Im(f) = Im(g). In addition, we provide in the Appendix a regulous and a regular map f, g : R 2 $\rightarrow$ R 2 whose common image is the open quadrant Q := {x > 0, y > 0}. These maps are much simpler than the best known polynomial maps R 2 $\rightarrow$ R 2 that have the open quadrant as their image.