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

Gianluca Gorni ; Gaetano Zampieri

Annales Polonici Mathematici, Tome 64 (1996) no. 3, pp. 285-290

Résumé

It is well known that the Jacobian conjecture follows if it is proved for the special polynomial mappings $f:ℂ^n → ℂ^n$ of the Yagzhev type: f(x) = x - G(x,x,x), where G is a trilinear form and $det f'(x) ≡ 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 $∑_{k≥1}Φ_k$ of homogeneous terms $Φ_k$ of degree k, we find that $Φ_{2m+1}$ is a linear combination of certain m-fold "nested compositions" of G with itself. If the Jacobian Conjecture were true, $f^{-1}$ should be a polynomial mapping of degree $≤ 3^{n-1}$ and the terms $Φ_k$ ought to vanish identically for $k > 3^{n-1}$. We may wonder whether the reason why $Φ_{2m+1}$ 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.

Détail

DOI : 10.4064/ap-64-3-285-290