Let \(F_ 1,...,F_ n\in {\mathbb{C}}[X_ 1,...,X_ n]\) be n polynomials, and let \(F=(F_ 1,...,F_ n):\quad {\mathbb{C}}^ n\to {\mathbb{C}}^ n\) be the corresponding polynomial transformation. If F has a polynomial inverse, then \(\det (\partial F_ i/\partial X_ j)\) is a nonzero constant. The Jacobian conjecture claims, that the opposite should be true. That conjecture has been reduced to the case, in which \(F_ i=X_ i+H_ i\), where \(H_ i\) are cubic forms and the jacobian \((\partial H_ i/\partial X_ j)\) is nilpotent [cf. \textit{H. Bass}, \textit{E. H. Connell} and \textit{D. Wright}, Bull. Am. Math. Soc., New Ser. 7, 287-330 (1982; Zbl 0539.13012)]. The author proves the Jacobian conjecture in the last formulation in the special case, when \((\partial H_ i/\partial X_ j)^ 2=0\).