Picturing Pinchuk's Plane Polynomial Pair

TL;DR

This paper analyzes the asymptotic behavior of a specific polynomial map from Pinchuk's class, providing detailed geometric visualization and correcting previous inaccuracies about its image.

Contribution

It offers a detailed geometric analysis of a Pinchuk polynomial map, correcting earlier errors and applying Peretz's techniques for asymptotic variety description.

Findings

Corrected the description of the map's image, showing it misses only two points.

Provided a detailed geometric visualization of the polynomial map.

Verified the asymptotic variety using advanced techniques.

Abstract

Sergey Pinchuk discovered a class of pairs of real polynomials in two variables that have a nowhere vanishing Jacobian determinant and define maps of the real plane to itself that are not one-to-one. This paper describes the asymptotic behavior of one specific map in that class. The level of detail presented permits a good geometric visualization of the map. Errors in an earlier description of the image of the map are corrected (the complement of the image consists of two, not four, points). Techniques due to Ronen Peretz are used to verify the description of the asymptotic variety of the map.