A counterexample to a conjecture of Bass, Connell and Wright

Let F=X-H:${\displaystyle k^n}$ → ${\displaystyle k^n}$ be a polynomial map with H homogeneous of degree 3 and nilpotent Jacobian matrix J(H). Let G=(G_1,...,G_n) be the formal inverse of F. Bass, Connell and Wright proved in [1] that the homogeneous component of ${\displaystyle G_i}$ of degree 2d+1 can be expressed as ${\displaystyle G_i^{\left(d\right)}=\sum _T\alpha {\left(T\right)}^{-1}{\sigma }_i\left(T\right)}$, where T varies over rooted trees with d vertices, α(T)=CardAut(T) and ${\displaystyle {\sigma }_i\left(T\right)}$ is a polynomial defined by (1) below. The Jacobian Conjecture states that, in our situation, ${\displaystyle F}$ is an automorphism or, equivalently, ${\displaystyle G_i^{\left(d\right)}}$ is zero for sufficiently large d. Bass, Connell and Wright conjecture that not only ${\displaystyle G_i^{\left(d\right)}}$ but also the polynomials ${\displaystyle {\sigma }_i\left(T\right)}$ are zero for large d. The aim of the paper is to show that for the polynomial automorphism (4) and rooted trees (3), the polynomial ${\displaystyle {\sigma }_2\left(T_s\right)}$ is non-zero for any index ${\displaystyle s}$ (Proposition 4), yielding a counterexample to the above conjecture (see Theorem 5).