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.
Balreira, E.C. (2018). A geometric generalization of the Planar Gale-Nikaidô theorem. Journal of Mathematics and Statistics, 14, 151-155. doi: 10.3844/jmssp.2018.151.155
Journal of Mathematics and Statistics
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