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 homogeneouscomponent 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 rootedtrees with $d$ vertices, $\alpha(T) = {\text{Card\,Aut}}(T)$ and, foreach 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 $f\colonV(T)\rightarrow\{1,\cdots ,n\}$ such that $f({\rm rt}_{T}) =i$). In thissituation, the Jacobian conjecture states that $F$ is an automorphismor, equivalently, $G_{i}^{(d)}$ is zero for all sufficiently large$d$. Bass, Connell and Wright [op. cit.] conjectured that not onlythe $G_{i}^{(d)}$ but also the $\sigma_{i}(T)$ are zero for all largeenough $d$. 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 acertain sequence of rooted trees $\{T_{s}\}$, the polynomial$\sigma_{2}(T_{s})$ is nonzero for every index $s$: Indeed, he showsthat $\sigma_{2}(T_{s}) = (-1)^{s+1}6X_{4}^{4s+7}(X_{1}X_{3}+X_{2}X_{4})$ for all $s\geq 0$. This providesa counterexample to the latter conjecture. As the author remarks, thisparticular endomorphism $F\colon {\bold C}^{4} \rightarrow {\boldC}^{4}$ was first constructed by Arno\,van\,den\,Essen in September1994 as a counterexample to a conjecture which the reviewer had madeat the July 1994 Curaçao Conference on Polynomial Maps [seeG. 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-homogeneouspolymorphism whose Schröder function $h_{s}$ is not itself apolynomial map, van den Essen won the$\$100$ prize I had offered at the Curaçao Conference. For a fulleraccount of these matters, and their serendipitous connection to thebeautiful final solution of the celebrated Markus-Yamabe conjecture,see the forthcoming survey paper by the reviewer [``A biography of theMarkus-Yamabe conjecture'', in {\it Proceedings of the Conference onAspects of Mathematics---Algebra, Geometry and Several ComplexVariables (Hong Kong, 1996)}, to appear]. It is nice to see that myinvestment of $\$100$ is still paying off! What more can be done withthis example?