A counterexample to a conjecture of Wright on homogeneous polynomial maps associated with rooted trees

Let F=X−H :kn→kn be a polynomial map such that H is homogeneous of degree d 3 and the Jacobian matrix of H is nilpotent. Homogeneous components of the formal inverse of F in the form given by Bass, Connell and Wright are linear combinations of polynomial maps σ(T) indexed by rooted trees. In later paper, Wright conjectures that σ(T)=0 if T is a rooted tree whose leaves have height n−1. We show that conjecture is false if n 5 and d 3.