A short proof for the open quadrant problem

TL;DR

This paper presents a concise, computer-free proof that the open quadrant in ^2 is a polynomial image of ^2, simplifying the original proof by decomposing the map into three understandable polynomial maps.

Contribution

It offers a new, simplified proof of a known result, avoiding computational methods and using a composition of three simple polynomial maps.

Findings

The open quadrant is a polynomial image of ^2.

The proof avoids computer calculations, making it more accessible.

The approach simplifies understanding polynomial images of Euclidean spaces.

Abstract

In 2003 it was proved that the open quadrant $\{x>0,y>0\}$ of ${\mathbb R}^2$ is a polynomial image of ${\mathbb R}^2$. 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.