A counterexample to a conjecture of Bass, Connell and Wright
Volume 77 / 1998
Abstract
Let F=X-H:$k^n$ → $k^n$ 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 $G_i$ of degree 2d+1 can be expressed as $G_i^{(d)}=\sum_T α(T)^{-1} σ_i(T)$, where T varies over rooted trees with d vertices, α(T)=CardAut(T) and $σ_i(T)$ is a polynomial defined by (1) below. The Jacobian Conjecture states that, in our situation, $F$ is an automorphism or, equivalently, $G_i^{(d)}$ is zero for sufficiently large d. Bass, Connell and Wright conjecture that not only $G_i^{(d)}$ but also the polynomials $σ_i(T)$ 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 $σ_2(T_s)$ is non-zero for any index $s$ (Proposition 4), yielding a counterexample to the above conjecture (see Theorem 5).
