Polynomial maps and a conjecture of Samuelson
It is shown that, for a polynomial map \(F: \mathbb{C}^ n\to \mathbb{C}^ n\), if the upper left-hand principal minors of all orders in the Jacobian matrix \(JF\) are non-zero at all points, then \(F\) has an inverse polynomial map. The authors first note that the non-vanishing of the minors implies that the minors are non-zero elements of \(\mathbb{C}\), as \(\mathbb{C}\) is algebraically closed. Then by induction on \(n\) they show that \(F\) is the product of a diagonal map of constants with \(n\) elementary polynomial maps (\(F\) is elementary of type \(i\) when \(F(x_ 1,\dots,x_ n)=(x_ 1,\dots,F_ 2(x_ 1,\dots,x_ n),\dots,x_ n)\) with \(F_ i(X_ 1,\dots,X_ n)- X_ i\) a polynomial in \(X_ 1,\dots,\hat X_ i,\dots,X_ n)\).
scientific article; zbMATH DE number 166226
scientific article