[Paper Review] Picturing Pinchuk's Plane Polynomial Pair

L. Andrew Campbell|arXiv (Cornell University)|Dec 5, 1998

Advanced Differential Equations and Dynamical Systems4 references3 citations

TL;DR

This paper provides a detailed geometric visualization of a specific polynomial map discovered by Sergey Pinchuk, demonstrating that despite having a nowhere vanishing Jacobian determinant, the map is not injective. Using techniques from Ronen Peretz, the author corrects a prior error, proving the complement of the image consists of exactly two points, and analyzes the asymptotic behavior to clarify the map's structure.

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.

Motivation & Objective

• To provide a geometric understanding of a specific polynomial map in the class discovered by Pinchuk that has a nowhere vanishing Jacobian but is not injective.
• To correct an earlier error in the description of the image's complement, which was incorrectly stated to consist of four points.
• To analyze the asymptotic behavior of the map using advanced algebraic geometry techniques.
• To offer a detailed visualization of the map's structure through asymptotic variety analysis.

Proposed method

• Application of techniques developed by Ronen Peretz to verify the asymptotic variety of the map.
• Use of algebraic geometry tools to study the image and complement of the polynomial map.
• Analysis of level sets and asymptotic behavior to visualize the map's global structure.
• Use of TeX and EPS files to generate precise geometric representations of the map.
• Comparison with prior work to identify and correct an error in the image description.
• Systematic examination of the map's behavior at infinity to determine the topology of its image complement.

Experimental results

Research questions

• RQ1 What is the true topological structure of the complement of the image of Pinchuk's polynomial map?
• RQ2 How does the asymptotic behavior of the map influence its global injectivity despite a non-vanishing Jacobian?
• RQ3 Why did earlier analyses incorrectly identify four points in the complement of the image?
• RQ4 What role does the asymptotic variety play in determining the map's non-injective nature?
• RQ5 How can geometric visualization be systematically achieved for such complex polynomial maps?

Key findings

• The complement of the image of the map consists of exactly two points, correcting a prior claim of four points.
• The asymptotic variety of the map is fully characterized using Peretz's techniques, confirming the geometric structure.
• The map exhibits non-injective behavior despite having a nowhere vanishing Jacobian determinant.
• The geometric visualization reveals a complex, non-trivial image structure with two missing points.
• The analysis confirms that the map is not injective, resolving a long-standing question about such polynomial pairs.
• The detailed asymptotic analysis provides a complete picture of the map's global behavior in the real plane.