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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:Rn→Rm is regulous if it is a rational map that admits a continuous extension to Rn⁠. In case the set of (real) poles of f is empty we say that it is regular map. We prove that if S⊂Rm is the image of a regulous map f:Rn→Rm⁠, there exists a dense semialgebraic subset T⊂S and a regular map g:Rn→Rm such that g(Rn)=T⁠. In case dim(S)=n⁠, we may assume that the difference S⧹T has codimension ≥2 in S⁠. If we restrict our scope to regulous maps from the plane the result is neat: iff:R2→Rmis a regulous map, there exists a regular mapg:R2→Rmsuch thatIm(f)=Im(g). In addition, we provide in Appendix A a regulous and a regular mapf,g:R2→R2whose common image is the open quadrantQ≔{𝚡>0,𝚢>0}. These maps are much simpler than the best-known polynomial mapsR2→R2that have the open quadrant as their image.

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