Pinchuk maps, function fields, and real Jacobian conjectures

TL;DR

This paper investigates the properties of rational maps in real n-space related to the Jacobian conjecture, focusing on function field extensions, automorphisms, and specific counterexamples, providing new insights and reduction techniques.

Contribution

It establishes conditions on function field extensions for real Jacobian conjectures, analyzes Pinchuk counterexamples, and proves reduction theorems for specialized forms.

Findings

Extensions for Pinchuk counterexamples have degree six and no nontrivial automorphisms.

The birational case of the conjecture is proved.

Certain topological conditions are sufficient for invertibility.

Abstract

Jacobian conjectures (that nonsingular implies invertible) for rational everywhere defined maps of real n-space to itself are considered, with no requirement for a constant Jacobian determinant or a rational inverse. The associated extension of rational function fields must be of odd degree and must have no nontrivial automorphisms. The extensions for the Pinchuk counterexamples to the strong real Jacobian conjecture have no nontrivial automorphisms, but are of degree six. The birational case is proved, the Galois case is clarified but the general case of odd degree remains open. However, certain topological conditions are shown to be sufficient. Reduction theorems to specialized forms are proved.