Yagzhev polynomial mappings: on the structure of the Taylor expansion of their local inverse

It is well known that the Jacobian conjecture follows if it is proved for the special polynomial mappings ${\displaystyle f:{ℂ}^n\to {ℂ}^n}$ of the Yagzhev type: f(x) = x - G(x,x,x), where G is a trilinear form and ${\displaystyle {\det f}^{\prime }\left(x\right)\equiv 1}$. Drużkowski and Rusek [7] showed that if we take the local inverse of f at the origin and expand it into a Taylor series ${\displaystyle {\sum }_{k\ge 1}{\Phi }_k}$ of homogeneous terms ${\displaystyle {\Phi }_k}$ of degree k, we find that ${\displaystyle {\Phi }_{2m+1}}$ is a linear combination of certain m-fold "nested compositions" of G with itself. If the Jacobian Conjecture were true, ${\displaystyle f^{-1}}$ should be a polynomial mapping of degree ${\displaystyle \le 3^{n-1}}$ and the terms ${\displaystyle {\Phi }_k}$ ought to vanish identically for ${\displaystyle k>3^{n-1}}$. We may wonder whether the reason why ${\displaystyle {\Phi }_{2m+1}}$ vanishes is that each of the nested compositions is somehow zero for large m. In this note we show that this is not at all the case, using a polynomial mapping found by van den Essen for other purposes.