On univalence and P-matrices

AbstractSuppose F is a differentiable mapping from a rectangle R⊂En into En. Gale and Nikaido proved that if the Jacobian of F is a P-matrix in R, then F is univalent in R. Their paper has served as the basis of numerous results on univalence. Recently H. Scarf conjectured a significant extension: that the Jacobian of F need not be a P-matrix everywhere in the rectangle R, but merely on its boundary. This paper proves Scarf's conjecture, and to do so utilizes a conceptually different method of proof than that of Gale and Nikaido. The proof is presented in such a way as to demonstrate a suggestion of Scarf that orientation arguments may provide an alternative proof of the Gale-Nikaido theorem