On the Rational Real Jacobian Conjecture

2012/09/30 by L. Andrew Campbell, Campbell, L. Andrew · 2 citations
Mathematics · Physics and Astronomy · #14R15 #Advanced Differential Equations and Dynamical Systems #Advanced Differential Geometry Research #Algebraic Geometry (math.AG) #FOS: Mathematics #Quantum chaos and dynamical systems #math.AG #msc:14R15

paper · pdf · doi:10.48550/arxiv.1210.0251

7 pages. arXiv admin note: substantial text overlap with arXiv:1202.2949

openalex publication_date 2012/09/30 · arxiv created 2013/11/15 · arxiv updated 2013/11/18 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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.

Citations

• A counterexamle to the strong real Jacobian conjecture
• Pinchuk maps, function fields, and real Jacobian conjectures

Cited by

• On quartic forms associated with cubic transformations of the real plane
• A rough classification of potentially invertible cubic transformations of the real plane

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