Research Article Open Access

A Geometric Generalization of the Planar Gale-Nikaidô Theorem

E. Cabral Balreira1
  • 1 Trinity University, United States

Abstract

The Gale-Nikaidô Theorem establishes global injectivity of maps defined over rectangular regions provided the Jacobian matrix is a P-matrix. We provide a purely geometric generalization of this result in the plane by showing that if the image of each edge of the rectangular domain is realized as a graph of a function over the appropriate axis, then the map is injective. We also show that the hypothesis that the Jacobian matrix is a P-matrix is simply one way to analytically check this geometric condition.

Journal of Mathematics and Statistics
Volume 14 No. 1, 2018, 151-155

DOI: https://doi.org/10.3844/jmssp.2018.151.155

Submitted On: 24 April 2018 Published On: 30 May 2018

How to Cite: Balreira, E. C. (2018). A Geometric Generalization of the Planar Gale-Nikaidô Theorem. Journal of Mathematics and Statistics, 14(1), 151-155. https://doi.org/10.3844/jmssp.2018.151.155

  • 3,958 Views
  • 1,871 Downloads
  • 0 Citations

Download

Keywords

  • Global Injectivity
  • P-Matrix
  • Gale-Nikaidô