On polynomial submersions of degree $4$ and the real Jacobian conjecture in ${\mathbb{R}}^2$

Abstract

The main result of this paper is the following version of the real Jacobian conjecture: "Let $F=(p,q):{\mathbb{R}}^2\to {\mathbb{R}}^2$ be a polynomial map with nowhere zero Jacobian determinant. If the degree of $p$ is less than or equal to $4$, then $F$ is injective". Assume that two polynomial maps from ${\mathbb{R}}^2$ to $\mathbb{R}$ are equivalent when they are the same up to affine changes of coordinates in the source and in the target. We completely classify the polynomial submersions of degree $4$ with at least one disconnected level set up to this equivalence, obtaining four classes. Then, analyzing the half-Reeb components of the foliation induced by a representative $p$ of each of these classes, we prove there is not a polynomial $q$ such that the Jacobian determinant of the map $(p,q)$ is nowhere zero. Recalling that the real Jacobian conjecture is true for maps $F=(p,q)$ when all the level sets of $p$ are connected, we conclude the proof of the main result.

References in corpus (1)

• Invertible polynomial mappings via Newton non-degeneracy

Cited by in corpus (3)

• Foliations, solvability and global injectivity
• Real Jacobian mates
• Jacobian conjecture in ${\mathbb{R}}^2$