Polynomial similarity of pairs of matrices

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 $A_2=X^{-1}{A}_{1}X$ and $B_2=X^{-1}{B}_{1}X$ 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 $\det J(f,g)(0,0)\ne 0,$ 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 $\tilde{P}(\tilde{A},\tilde{B})$ 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 $\overline{P}$, $\tilde{P}$ 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.