Rational Samuelson maps are univalent
Let \(F = (f_ 1, \dots, f_ n) : \mathbb{R}^ n \to \mathbb{R}^ n\) be a differentiable map. \(F\) is called a Samuelson map iff all determinants \[ \mu_ r := \text{det}\left( {\partial f_ i \over \partial x_ j} \right)_{\begin{smallmatrix} 1 \leq i \leq r\\ 1 \leq j \leq r\end{smallmatrix}},\quad r = 1,\dots, n, \] are everywhere nonzero. The author proves the following theorem: Let \(F\) be a Samuelson \(C^ 1\)-map. If all functions \(f_ i\) have an algebraic graph then \(F\) is bijective. The author also deals with related questions, for instance the Jacobian conjecture. He gives many examples. Actually he proves stronger versions of the theorem.
scientific article; zbMATH DE number 540915
scientific article