The open quadrant problem: A topological proof

Abstract

In this work we present a new polynomial map $f:=(f_1,f_2):{\mathbb R}^2\to{\mathbb R}^2$ whose image is the open quadrant $\{x>0,y>0\}\subset{\mathbb R}^2$. The proof of this fact involves arguments of topological nature that avoid hard computer calculations. In addition each polynomial $f_i\in{\mathbb R}[{\tt x},{\tt y}]$ has degree $\leq16$ and only $11$ monomials, becoming the simplest known map solving the open quadrant problem.