Triangularization properties of power linear maps and the structural conjecture (Q2925544)

The Jacobian conjecture, which asks whether every polynomial map \(F: K^n \to K^n\) over a field \(K\) of characteristic zero and \(\det DF \in K^*\) must be invertible, remains open for all \(n \geq 2\). \textit{L. M. Drużkowski} [Ann. Pol. Math. 41, 157--165 (1983; Zbl 0569.32001)] reduced the problem to \textit{cubic power linear maps} \(F(x) = x + \sum_{k=1}^n l_j(x)^3 e_j\), where \(l_j(x)\) are linear forms in \(x\). For maps \(F\) of this form satisfying Keller's hypothesis \(\det DF = 1\), he conjectured that \(\det (DF|_{x=v_1} + DF|_{x=v_2}) \neq 0\) for all \(v_1, v_2 \in \mathbb C^n\) iff there exist \(b_j, c_j \in \mathbb C^n\) for which \(F(x) = x+\sum_{i=1}^{n-1} (c_i^t x)^3 b_i\).NEWLINENEWLINEThe authors study this conjecture for maps of the form \(F(x) = x + H\) with \(H\) a degree \(d\) polynomial. Using earlier work of \textit{H. Guo} et al. [Indag. Math., New Ser. 23, No. 3, 256--268 (2012; Zbl 1376.14062)], they prove easily the implications \((a) \Rightarrow (b) \Rightarrow (c) \Rightarrow (d) \Rightarrow (e) \Rightarrow (f)\) for the statements {\parindent=6mm \begin{itemize}\item[(a)] \(H=\sum_{i=1}^{n-1} (c_i^t x)^{d_i} b_i\) with \(1 \leq d_i \leq d\) and \(b_i\) linearly independent. \item[(b)] \(H=\sum_{i=1}^{n-1} (c_i^t x)^{d_i} b_i\) with \(1 \leq d_i \leq d\). \item[(c)] \(H=\sum_{i=1}^{N} (c_i^t x)^{d_i} b_i\) for some \(N > 0\). \item[(d)] \(\det(\sum_{i=1}^n DF|_{x=v_i}) \in {\overline K}^*\) for all \(v_i \in {\overline K}^n\). \item[(e)] \(\det(\sum_{i=1}^{d-1} DF|_{x=v_i}) \in {\overline K}^*\) for all \(v_i \in {\overline K}^n\). \item[(f)] \(F\) is an invertible map. NEWLINENEWLINE\end{itemize}} The first three conditions are equivalent to various forms of linear triangulizability. The interesting point is that most of the converses fail, even for homogeneous power linear maps (i.e. \(d_i = d\) constant), namely \((f) \not \Rightarrow (e), (d) \not \Rightarrow (c) \not \Rightarrow (b) \not \Rightarrow (a)\). Taking \(d=3\) proves one direction of Drużkowski's conjecture, but the examples for \((d) \not \Rightarrow (c) \not \Rightarrow (b)\) disproves the other direction and shows that condition \((c)\), which is equivalent to \(H\) being linearly triangularizable with nilpotent derivative matrix \(DH\), is strictly in between. The authors also give some positive results for small values of \(d\) and \(n\).