부산대학교 도서관

Let $F=X-H\colon k^{n}\rightarrow k^{n}$ be a polynomial map with $H$ homogeneous of degree 3 and nilpotent Jacobian matrix $J(H)$. Let $G = (G_{1}, \cdots ,G_{n})$ be the formal inverse of $F$. H. Bass, E. H. Connell and D. Wright\ [Bull. Amer. Math. Soc. (N.S.) {\bf 7} (1982), no.~2, 287--330; MR0663785 (83k:14028)] proved that the homogeneous component of $G_{i}$ of degree $2d+1$ can be expressed as $G_{i}^{(d)} = \sum_{T}\alpha(T)^{-1}\sigma_{i}(T)$, where $T$ varies over rooted trees with $d$ vertices, $\alpha(T) = {\text{Card\,Aut}}(T)$ and, for each fixed rooted tree $T$ and integer $i\in \{1, \cdots ,n\}$, $\sigma_{i}(T)$ is the polynomial defined by $\sigma_{i}(T) \coloneq \sum_{f}\prod_{v\in V(T)}((\prod_{w\in v^{+}}D_{f(w)})H_{f(v)})$, $f$ varying over all $i$-rooted labelings of $T$ (functions $f\colon V(T)\rightarrow\{1,\cdots ,n\}$ such that $f({\rm rt}_{T}) =i$). In this situation, the Jacobian conjecture states that $F$ is an automorphism or, equivalently, $G_{i}^{(d)}$ is zero for all sufficiently large $d$. Bass, Connell and Wright [op.\ cit.] conjectured that not only the $G_{i}^{(d)}$ but also the $\sigma_{i}(T)$ are zero for all large enough $d$. The author shows that for the polynomial automorphism $F = (X_{1}+X_{4}(X_{1}X_{3}+X_{2}X_{4}), X_{2}-X_{3}(X_{1}X_{3}+X_{2}X_{4}), X_{3}+X_{4}^{3},X_{4})$, and for a certain sequence of rooted trees $\{T_{s}\}$, the polynomial $\sigma_{2}(T_{s})$ is nonzero for every index $s$: Indeed, he shows that $\sigma_{2}(T_{s}) = (-1)^{s+1} 6X_{4}^{4s+7}(X_{1}X_{3}+X_{2}X_{4})$ for all $s\geq 0$. This provides a counterexample to the latter conjecture. As the author remarks, this particular endomorphism $F\colon {\bold C}^{4} \rightarrow {\bold C}^{4}$ was first constructed by Arno\,van\,den\,Essen in September 1994 as a counterexample to a conjecture which the reviewer had made at the July 1994 Cura\c{c}ao Conference on Polynomial Maps [see G. H. Meisters, in {\it Automorphisms of affine spaces (Curaçao, 1994)}, 67--87, Kluwer Acad. Publ., Dordrecht, 1995; MR1352691 (97m:14020); A. van den Essen, ibid., 231--233; MR1352704 (96h:14022)]. With this example of a cubic-homogeneous polymorphism whose Schröder function $h_{s}$ is not itself a polynomial map, van den Essen won the $\$100$ prize I had offered at the Cura\c{c}ao Conference. For a fuller account of these matters, and their serendipitous connection to the beautiful final solution of the celebrated Markus-Yamabe conjecture, see the forthcoming survey paper by the reviewer [``A biography of the Markus-Yamabe conjecture'', in {\it Proceedings of the Conference on Aspects of Mathematics---Algebra, Geometry and Several Complex Variables (Hong Kong, 1996)}, to appear]. It is nice to see that my investment of $\$100$ is still paying off! What more can be done with this example? $