Open Problems in Integrable Systems — Nijenhuis operators: singular points and global properties

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

For $\sigma=(\sigma_1,\ldots,\sigma_n)$, define the companion matrix $C(\sigma)$ by \[ C_{i1}=-\sigma_i\quad(1\leq i\leq n),\qquad C_{i,i+1}=1\quad(1\leq i<n), \] with all remaining entries zero. For a map $\Phi=(\sigma_1,\ldots,\sigma_n)$ in coordinates $x_1,\ldots,x_n$, its Jacobian is $J_{ij}=\partial\sigma_i/\partial x_j$.

Let $\sigma_i\in\mathbb R[x_1,\ldots,x_n]$ be homogeneous of degree $i$ for $1\leq i\leq n$, and assume they are algebraically independent. Then $D=\det J$ is a nonzero polynomial and \[ L=\frac{\operatorname{adj}(J)C(\sigma)J}{D}. \] Each entry is a homogeneous rational function of degree one. Write $Q_{ij}$ for the corresponding numerator in $\operatorname{adj}(J)C(\sigma)J$.

Statement recorded in the supplied source.

Classify all such collections $(\sigma_1,\ldots,\sigma_n)$ for which $D$ divides every $Q_{ij}$ in $\mathbb R[x_1,\ldots,x_n]$. Equivalently, require every entry of $L$ to be a homogeneous linear polynomial in $x_1,\ldots,x_n$. The reconstructed field then has zero Nijenhuis torsion and corresponds to a left-symmetric algebra, as described in the source. [S2]

Short English statement

Classify algebraically independent homogeneous coefficient polynomials for which the reconstructed tensor has linear polynomial entries.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2100513 (AMR-020-0513). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Alexey Bolsinov, Vladimir S. Matveev, Eva Miranda and Serge Tabachnikov, Open Problems, Questions, and Challenges in Finite-Dimensional Integrable Systems (2018), Problem 5.16 and its complete preceding definitions. Original question and full discussion. https://arxiv.org/abs/1804.03737.