On regulous and regular images of Euclidean spaces

TL;DR

This paper compares regulous and regular maps from Euclidean spaces, showing that regulous images can be closely approximated by regular images, especially in the plane, and provides explicit examples of maps with the open quadrant as their image.

Contribution

It establishes a connection between regulous and regular images, demonstrating that regulous images can be approximated by regular images with high codimension differences, and offers explicit simple maps for the open quadrant.

Findings

Regulous images can be densely approximated by regular images.

In the plane, regulous and regular images coincide.

Explicit simple maps for the open quadrant are constructed.

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.