A new class of non-injective polynomial local diffeomorphisms on the plane

Abstract

In this short note we provide the example with the lowest degree known so far of a non-injective local polynomial diffeomorphism $F=(p,q):\mathbb{R}^2 \to \mathbb{R}^2$. In our example $p$ has degree $10$ and $q$ has degree $15$, rather than $10$ and $25$, respectively, known up to now as the smallest degrees for the coordinates of $F$. Our construction was based on S. Pinchuk celebrated counterexample to the real Jacobian conjecture.