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 $A_2=X^{-1}A_1X$ 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.