F. Braun | Searcharxiv

arXiv2026

A sharp degree bound in the real Jacobian conjecture

Let $F=(p,q):\mathbb R^2\to \mathbb R^2$ be a polynomial map with nowhere zero Jacobian determinant. A long-standing problem is to determine the largest integer $k$ such that the condition $\deg p\le k$ guarantees the global injectivity of $F$. Although several partial results have been obtained over the past $30$ years, the sharp degree bound has remained unknown. In this paper, we prove that $F$ is injective whenever $\deg p=6$. On the other hand, we construct a non-injective polynomial map with nowhere vanishing Jacobian determinant for which $\deg p=7$. Combined with the previously known injectivity results for $\deg p\le 5$, our results completely settle the problem and establish the optimal degree bound. More precisely, we show that $7$ is the minimal degree for which non-injective examples can occur.

math.AG↗

arXiv2014

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

The main result of this paper is the following version of the real Jacobian conjecture: "Let $F=(p,q):\R^2\to\R^2$ 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 $\R^2$ to $\R$ 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.

math.DS↗