A counterexample to a conjecture of Wright on homogeneous polynomial maps associated with rooted trees
Let \(K\) be a field and \(F_1,\dots, F_n\in K[x]\), \(X=(X_1, \dots, X_n)\), \(F=(F_1, \dots, F_n)\). Let \(J(F)=({\partial F_i\over \partial X_j})\) be the Jacobian matrix and assume that \(\det(J(F))\in K\subset\{0\}\). The Jacobian conjecture states that there exists a polynomial map \(G:K^n\to K^n\) such that \(G(F)=X\). There is a conjecture of \textit{D. Wright} [J. Pure Appl. Algebra 48, 199--219 (1987; Zbl 0666.12017)] concerning the case \(F=X-H\), \(H\) homogeneous of degree \(d\geq 3\) and the Jacobian matrix \(J(H)\) being nilpotent, which implies for \(d=3\) the Jacobian conjecture. It is proven that Wright's conjecture is false for \(d\geq 3\) and \(n\geq 5\).
scientific article; zbMATH DE number 1930185