The Jacobian conjecture in two variables (Q1061790)

The authors prove a special case of the Jacobian conjecture: if \(F=(f,g): {\mathbb{C}}^ 2\to {\mathbb{C}}^ 2\) is a polynomial map whose Jacobian determinant is a nonzero constant, then F is invertible, provided the degree of f or g is a product of at most two primes. The proof uses and extends ideas of \textit{Y. Nakai} and \textit{K. Baba} concerning weighted graduations on the polynomial ring \({\mathbb{C}}[X,Y]\) [cf. Osaka J. Math. 14, 403-409 (1977; Zbl 0373.12010)]. \{Reviewer's remark: Part of the Lemmas used in the proof can also be found (with different proofs) in \textit{S. S. Abhyankar}'s Tata Lect. Notes 57 (''On expansion techniques in algebraic geometry'', Bombay 1977).\}