The PInchuk example revisited

TL;DR

This paper revisits specific examples of non-injective polynomial maps from  to  with non-zero Jacobian, highlighting their unique properties such as surjectivity and non-dense images.

Contribution

It provides detailed analysis of two special polynomial maps with non-vanishing Jacobian, illustrating their distinct geometric behaviors.

Findings

One example is surjective.

Another example has a non-dense image.

Both examples have non-injective polynomial maps with non-zero Jacobian.

Abstract

In this note we provide two special examples of non-injective polynomial maps from $\mathbb{R}^2$ to $\mathbb{R}^2$ with non-vanishing Jacobian: the first one is surjective, the second one has non-dense image.