A Geometric Generalization of the Planar Gale-Nikaidô Theorem
E. Cabral Balreira
DOI : 10.3844/jmssp.2018.151.155
Journal of Mathematics and Statistics
Volume 14, 2018
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.
© 2018 E. Cabral Balreira. This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.