A short proof for the open quadrant problem

Abstract

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.

On regulous and regular images of Euclidean spaces

“…Finally in Appendix A we propose a regulous and a regular map f, g : R 2 → R 2 whose common image is the open quadrant Q := {x > 0, y > 0}. Theses maps are much more simpler than the best known polynomial maps R 2 → R 2 that have Q as their image [FG1,FU4,FGU3]. In addition, the verification that the images of f, g is Q is quite straightforward and does not require the enormous effort needed for the known polynomial maps used in [FG1,FU4,FGU3] to represent Q.…”

“…Theses maps are much more simpler than the best known polynomial maps R 2 → R 2 that have Q as their image [FG1,FU4,FGU3]. In addition, the verification that the images of f, g is Q is quite straightforward and does not require the enormous effort needed for the known polynomial maps used in [FG1,FU4,FGU3] to represent Q. Recall that the systematic study of the polynomial and regular images began in 2005 with the first solution to the open quadrant problem [FG1].…”

On Nash images of Euclidean spaces

“…In [FG1,FG2] we presented the first step to approach Problem 1.1. In [Fe1] appears a complete solution to Problem 1.1 for the 1-dimensional case, whereas in [FGU1,FGU3,FU1,FU2,FU4,FU5,U1,U2] we approached constructive results concerning the representation as either polynomial or regular images of the semialgebraic sets with piecewise linear boundary commented above. A survey concerning this topic, which provides the reader a global idea of the state of the art, can be found in [FGU4].…”

The open quadrant problem: A topological proof

“…In fact he provided in his Ph.D. Thesis a different topological proof for the map proposed in [4], together with an algebraic proof involving a different polynomial map. This map has inspired the first and third authors for a short algebraic proof of the open quadrant problem involving a new polynomial map [11] and has led us to look for a polynomial map with optimal algebraic structure whose image is the open quadrant. It is important to establish clearly the meaning of 'optimal algebraic structure' [11, §3(A)].…”

“…The example in [4] has total degree 56 and its total number of monomials is 168. The polynomial map in [11] has total degree 72 and its total number of monomials is 350. In this work we will prove:…”