Polynomial similarity of pairs of matrices

TL;DR

This paper proves that classifying pairs of matrices up to polynomial similarity over a field is a complex problem that encompasses the classical unsolvable matrix similarity classification problem.

Contribution

It introduces the concept of polynomial similarity for matrix pairs and demonstrates that this classification problem is wild and as hard as the classical matrix similarity problem.

Findings

Polynomial similarity is a well-defined equivalence relation.

The classification problem for polynomial similarity is proven to be wild.

It contains the classical problem of classifying matrix pairs up to similarity.

Abstract

Let $K$ be a field, $R=K[x, y]$ the polynomial ring and $\mathcal{M}(K)$ the set of all pairs of square matrices of the same size over $K.$ Pairs $P_1=(A_1,B_1)$ and $P_2=(A_2,B_2)$ from $\mathcal{M}(K)$ are called similar if $A_2=X^{-1}A_1X$ and $B_2=X^{-1}B_1X$ for some invertible matrix $X$ over $K$. Denote by $\mathcal{N}(K)$ the subset of $\mathcal{M}(K)$, consisting of all pairs of commuting nilpotent matrices. A pair $P$ will be called {\it polynomially equivalent} to a pair $\overline{P}=(\overline{A}, \overline{B})$ if $\overline{A}=f(A,B), \overline{B}=g(A ,B)$ for some polynomials $f, g\in K[x,y]$ satisfying the next conditions: $f(0,0)=0, g(0,0)=0$ and $ {\rm det} J(f, g)(0, 0)\not =0,$ where $J(f, g)$ is the Jacobi matrix of polynomials $f(x, y)$ and $g(x, y).$ Further, pairs of matrices $P(A,B)$ and $\widetilde{P}(\widetilde{A}, \widetilde{B})$ from $\mathcal{N}(K)$ will be called {\it polynomially similar} if there exists a pair $\overline{P}(\overline{A}, \overline{B})$ from $\mathcal{N}(K)$ such that $P$, $\overline{P}$ are polynomially equivalent and $\overline{P}$, $\widetilde{P}$ are similar. The main result of the paper: it is proved that the problem of classifying pairs of matrices up to polynomial similarity is wild, i.e. it contains the classical unsolvable problem of classifying pairs of matrices up to similarity.