Quadratic linear Keller maps of nilpotency index three
Let \(f=x+H:\mathbb{C}^{n}\rightarrow \mathbb{C}^{n}\) be a polynomial map such that \(H\) is homogeneous of degree \(d\geq 2.\) Let \(F_{A}=y+(Ay)^{d}: \mathbb{C}^{n}\rightarrow \mathbb{C}^{n},\) \(A\in M_{n,n}(\mathbb{C}),\) be a power linear map such that \(\;f\) and \(F_{A}\) are a generalized Gorni-Zampieri pair i.e. there exist matrices \(B\in M_{n,N}(\mathbb{C})\) and \(C\in M_{N,n}(\mathbb{C})\) such that 1. \(f(x)=BF_{A}(C(x)),\) 2. \(BC=I_{n},\) 3. \(\ker B=\ker A.\) For a given \(f\) such a \(F_{A}\) always exists. The authors prove the following relations between \(f\) and \(F_{A}:\) 1. for indices of nipotency \[ \operatorname{Ind}\operatorname{Jac}(A(y))^{d}=\operatorname{Ind}\operatorname{Jac}H+1, \] 2. \(f\) is linearly triangularizable if and only if \(F_{A}\) is linearly triangularizable. As a consequence they obtain that quadratic linear Keller map \( F_{A}=y+(Ay)^{2}\) with nilpotency index three i.e. \(\left( \operatorname{Jac} (A(y))^{2}\right) ^{3}=0,\) is linearly triangularizable.
scientific article; zbMATH DE number 5285754
scientific article