Note on an elementary proof of a theorem of Nakai and Baba on the Jacobian conjecture in two variables (Q752089)

Let P, Q be two polynomial-functions in \({\mathbb{C}}[x,y]\). The two- dimensional Jacobian conjecture states that if \(\partial (P,Q)/\partial (x,y)\) is non-zero, then \({\mathbb{C}}[x,y]={\mathbb{C}}[P,Q]\). It has been proved under certain restrictive assumptions concerning the degrees of P and Q. In particular Nakai and Baba have shown that the conjecture is true if the degree of P or Q is 4 or if the larger degree is 2p \((p>2\) being prime). A direct proof by means of elementary computations only is given in this paper.