Real Jacobian mates

TL;DR

This paper investigates conditions under which certain real polynomials in two variables lack real Jacobian mates, contributing to the understanding of polynomial mappings with positive Jacobian determinants.

Contribution

It introduces a class of polynomials that do not admit real Jacobian mates, advancing the theory of polynomial mappings with positive Jacobian determinants.

Findings

Identifies a specific class of polynomials without real Jacobian mates

Provides criteria for the non-existence of real Jacobian mates

Enhances understanding of polynomial mappings in real algebraic geometry

Abstract

Let $p$ be a real polynomial in two variables. We say that a polynomial $q$ is a real Jacobian mate of $p$ if the Jacobian determinant of the mapping $(p,q):\mathbb{R}^2\to\mathbb{R}^2$ is everywhere positive. We present a class of polynomials that do not have real Jacobian mates.