부산대학교 도서관

Let $\bold C$ denote the field of complex numbers and let $F_1, \cdots,F_n$ be $n$ polynomials with complex coefficients in $n$ variables $X_1,\cdots,X_n$. Then $F=(F_1,\cdots,F_n)$ is a polynomial transformation of the affine $n$-space $\bold C^n$ into itself. Let $F'=(\partial F_i/\partial X_j)$ denote the Jacobian matrix of $F$. The Jacobian conjecture (of Otto-Heinrich Keller) is that $F$ has a polynomial inverse provided only that $\det F'$ is a nonzero constant (an obviously necessary condition). \n H. Bass, E. H. Connell and D. L. Wright\en [Bull. Amer. Math. Soc. (N.S.) {\bf 7} (1982), no. 2, 287--330; MR0663785 (83k:14028)] showed that the proof of the Jacobian conjecture can be reduced to the case in which each $F_i=X_i+H_i$, where $H_i$ is a homogeneous polynomial of degree $3$ and the Jacobian matrix (derivative) $H'$ is nilpotent. In the same paper [op.\ cit., see p. 329] they settled (in the affirmative) the case when $(H')^2=0$. Now, in the paper under review, the author gives another proof of this case and also shows that (in this case) $F$ is generated by ``elementary'' or ``Jonquières'' automorphisms. More precisely, he proves (under all of the above conditions) that there exists an invertible linear homogeneous transformation $G$ of $\bold C^n$ into itself such that $(G^{-1}\circ H\circ G)'$ is a triangular matrix. In particular, $F$ has a polynomial inverse.