[Paper Review] On the Rational Real Jacobian Conjecture

L. Andrew Campbell|arXiv (Cornell University)|Sep 30, 2012

Advanced Differential Equations and Dynamical Systems17 references3 citations

TL;DR

This paper investigates the Rational Real Jacobian Conjecture (RRJC), extending the Strong Real Jacobian Conjecture to everywhere-defined rational maps. It proves that for birational and Galois extensions, invertibility is equivalent to the field extension having odd degree and trivial automorphism group, and generalizes two known special cases of the SRJC to the rational map setting, showing invertibility when the set of non-trivial fibers has codimension >2 or is disjoint from the image.

ABSTRACT

Jacobian conjectures (that nonsingular implies a global inverse) 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 birational case is proved and the Galois case clarified. Two known special cases of the Strong Real Jacobian Conjecture (SRJC) are generalized to the rational map context. For an invertible map, the associated extension of rational function fields must be of odd degree and must have no nontrivial automorphisms. That disqualifies the Pinchuk counter examples to the SRJC as candidates for invertibility.

Motivation & Objective

• To extend the Strong Real Jacobian Conjecture (SRJC) to rational maps that are everywhere defined on ℝⁿ.
• To clarify conditions under which such rational maps are invertible, particularly focusing on field extension properties.
• To generalize two known special cases of the SRJC—concerning fiber sets of codimension >2 and disjointness from the image—into the rational map framework.
• To show that the Pinchuk counterexamples to the SRJC do not satisfy necessary conditions for invertibility under the RRJC.
• To establish that for invertible rational maps, the associated rational function field extension must be of odd degree and have trivial automorphism group.

Proposed method

• Analyzes the algebraic structure of the field extension ℝ(X)/ℝ(F) associated with a rational map F:ℝⁿ→ℝⁿ.
• Uses the degree of the field extension and the geometric degree of F to relate fiber size and extension parity.
• Applies topological methods, including covering space theory and P.A. Smith theory, to show that finite covering maps over ℝⁿ with nontrivial fundamental group lead to contradictions unless degree is 1.
• Leverages the fact that ℝⁿ is simply connected and a universal cover to deduce that a proper map with finite fibers must be injective if the covering degree is 1.
• Applies results from algebraic geometry and real algebraic geometry, particularly concerning semi-algebraic sets and dimension bounds on the set A(F) of non-trivial fibers.
• Uses the property that rational maps with nowhere vanishing Jacobian are locally diffeomorphic, and combines this with global injectivity criteria.

Experimental results

Research questions

• RQ1 Under what conditions is a rational, everywhere-defined map F:ℝⁿ→ℝⁿ with nowhere vanishing Jacobian determinant invertible?
• RQ2 How do the degree of the field extension ℝ(X)/ℝ(F) and the automorphism group of the extension relate to the invertibility of F?
• RQ3 Can the two known special cases of the SRJC—based on codimension of the set A(F) and its disjointness from the image—be generalized to rational maps?
• RQ4 Why do the Pinchuk counterexamples to the SRJC fail to satisfy the necessary conditions for invertibility under the RRJC?
• RQ5 What is the precise relationship between birationality, Galois extensions, and invertibility in the context of rational maps?

Key findings

• For a rational nonsingular map F:ℝⁿ→ℝⁿ, if the associated field extension ℝ(X)/ℝ(F) is birational (degree 1), then F is invertible and the extension has trivial automorphism group.
• If the field extension ℝ(X)/ℝ(F) is Galois, then F is invertible if and only if it is birational, which implies the extension degree is 1 and the automorphism group is trivial.
• The set A(F) of points over which F is not a locally trivial fibration must have codimension >2 or be disjoint from the image F(ℝⁿ) for F to be invertible, and this condition generalizes two known SRJC special cases to the RRJC setting.
• The Pinchuk counterexamples to the SRJC are disqualified as candidates for invertibility under the RRJC because they have fiber size 2, which is even, contradicting the necessary odd fiber size condition.
• Invertible rational maps must induce field extensions of odd degree and trivial automorphism group, and this condition rules out all known counterexamples to the SRJC.
• The proof of invertibility in the codimension >2 and disjointness cases relies on topological arguments involving covering spaces and Smith theory, and extends naturally to rational maps due to rationality ensuring finite covering degree and injectivity implying invertibility.