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.