The open quadrant problem: A topological proof

TL;DR

This paper introduces a new polynomial map from R^2 to R^2 that maps onto the open quadrant, using topological arguments to avoid complex computations, and presents the simplest known solution with degree ≤16 and 11 monomials.

Contribution

It provides the first topological proof of a polynomial map onto the open quadrant with minimal degree and monomials, simplifying previous solutions.

Findings

Polynomial map with degree ≤16 and 11 monomials maps onto the open quadrant.

Topological arguments can replace computational methods in proving such maps.

The constructed map is the simplest known solution to the open quadrant problem.

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.