Abstract. In this work we prove constructively that the complement R n \K of a convex polyhedron K ⊂ R n and the complement R n \ Int(K) of its interior are regular images of R n . If K is moreover bounded, we can assure that R n \ K and R n \ Int(K) are also polynomial images of R n . The construction of such regular and polynomial maps is done by double induction on the number of facets (faces of maximal dimension) and the dimension of K; the careful placing (first and second trimming positions) of the involved convex polyhedra which appear in each inductive step has interest by its own and it is the crucial part of our technique.

We show that convex polyhedra in R n and their interiors are images of regular maps R n → R n . As a main ingredient in the proof, given an n-dimensional, bounded, convex polyhedron K ⊂ R n and a point p ∈ R n \ K, we construct a semialgebraic partition {A, B, T} of the boundary ∂K of K determined by p, and compatible with the interiors of the faces of K, such that A and B are semialgebraically homeomorphic to an (n − 1)-dimensional open ball and T is semialgebraically homeomorphic to an (n − 2)-dimensional sphere. Finally, we also prove that closed balls in R n and their interiors are images of regular maps R n → R n .

Abstract. In this work we prove that the set of points at infinity S8 :" Cl RP m pSq X H8 of a semialgebraic set S Ă R m that is the image of a polynomial map f : R n Ñ R m is connected. This result is no longer true in general if f is a regular map. However, it still works for a large family of regular maps that we call quasi-polynomial maps.

Abstract. Let K ⊂ R n be a convex polyhedron of dimension n. Denote S := R n \ K and let S be its closure. We prove that for n = 3 the semialgebraic sets S and S are polynomial images of R 3 . The former techniques cannot be extended in general to represent the semialgebraic sets S and S as polynomial images of R n if n ≥ 4.

In 2003 it was proved that the open quadrant Q := {x > 0, y > 0} of R 2 is a polynomial image of R 2 . This result was the origin of an ulterior more systematic study of polynomial images of Euclidean spaces. In this article we provide a short proof of the previous fact that does not involve computer calculations, in contrast with the original one. The strategy here is to represent the open quadrant as the image of a polynomial map that can be expressed as the composition of three simple polynomial maps whose images can be easily understood.