Almost Differentially Nondegenerate Nijenhuis Operators
Dinmukhammed Akpan
TL;DR
The paper addresses the classification of Nijenhuis operators $L$ in $n$ dimensions near points where the first $n-1$ invariants of the characteristic polynomial are functionally independent and the determinant $\det L$ is arbitrary, formalized via $\chi(t)=\det(tI-L)=t^n+\sigma_1 t^{n-1}+\dots+\sigma_n$. It proves a general form (Theorem 1) by choosing coordinates so that $\sigma_1=x_1,\dots,\sigma_{n-1}=x_{n-1}$ and $\sigma_n=f$, yielding an explicit matrix realization of $L$ and reducing admissibility to the smoothness of the fractions $(\sum x_i f_{x_i}- f_{x_1} f_{x_{n-1}} - f)/f_y$ and $(f_{x_{j-1}}+ f_{x_j} f_{x_{n-1}})/f_y$. In the Morse-singularity case for $\sigma_n$, a coordinate change gives $f=\pm y^2$ and a further refined matrix form for $L$ with a $\pm 2y$ term; the special case $n=2$ admits an additional solution $R= x^2/4$. Overall, the results extend Akpan’s 2D findings to higher dimensions and provide explicit normal forms for integrability analyses and coordinate reductions.
Abstract
The paper is devoted to the study of Nijenhuis operators of arbitrary dimension $n$ in a neighborhood of a point at which the first $n-1$ coefficients of the characteristic polynomial are functionally independent, and the last coefficient (the determinant of the operator) is an arbitrary function. We prove a theorem on the general form of such Nijenhuis operators, and also obtain their complete description for the case in which the determinant has a nondegenerate singularity.