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 polynomialtransformation 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 nonzeroconstant (an obviously necessary condition). H. Bass, E. H. Connell andD. L. Wright [Bull. Amer. Math. Soc. (N.S.) {\bf 7} (1982),no. 2, 287--330; MR0663785 (83k:14028)] showed that the proof of the Jacobianconjecture 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 Jacobianmatrix (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 caseand also shows that (in this case) $F$ is generated by ``elementary''or ``Jonquières'' automorphisms. More precisely, he proves (underall of the above conditions) that there exists an invertible linearhomogeneous transformation $G$ of $\bold C^n$ into itself suchthat $(G^{-1}\circ H\circ G)'$ is a triangular matrix. In particular,$F$ has a polynomial inverse.