Analytic linearizability of some resonant vector fields (Q2719025)

The authors consider the holomorphic family of vector fields \(X(x,w)\), \(x\in U \subset \mathbb{C}^n\), \(w \in W \subset \mathbb{C}^m\), having a singular point at the origin of \(\mathbb{C}^n\). NEWLINE\[NEWLINEX(x,w)=A(w)x+a(x,w)NEWLINE\]NEWLINE where, the matrix \(A(w)\) denotes the linear part, and the vector \(a(x,w)\) the higher terms.NEWLINENEWLINENEWLINELet \(\lambda_1, \dots , \lambda_n\) be the eigenvalues of \(A(w)\). The nonlinearity \(a(x,w)\) is said to be admissible if there exists a constant \(\gamma \in \mathbb{C}\), such that \(a(x,w)\) is spanned by monomial vector fields \(\prod_{j=1}^n x_j^{\alpha_j}e_i\), where \(\sum_j \alpha_j \lambda_j - \lambda_i= k \gamma\), for some integer \(k\).NEWLINENEWLINENEWLINEThe authors prove, in two special cases, that there exists an analytic linearization of \(X(x,w)\):NEWLINENEWLINENEWLINE1. \(A(w)\) is diagonal and \(a(x,w)\) is admissible.NEWLINENEWLINENEWLINE2. \(A\) does not depend on \(w\), \(0 \not\in \text{Conv}(\lambda_1, \dots , \lambda_n)\) and for sufficiently large \(d\) the \(d\)-jet of \(a\) is admissible.