2024/08/08 by Bondarenko, Vitaliy, Petravchuk, Anatoliy, Styopochkina, Maryna
#15A21 #15A99 #16G60 #FOS: Mathematics #Representation Theory (math.RT)
paper · doi:10.48550/arxiv.2408.04244
Let K be a field, R=K[x, y] the polynomial ring and M(K) the set of all pairs of square matrices of the same size over K. Pairs P1=(A1,B1) and P2=(A2,B2) from M(K) are called similar if A2=X-1A1X and B2=X-1B1X for some invertible matrix X over K. Denote by N(K) the subset of M(K), consisting of all pairs of commuting nilpotent matrices. A pair P will be called \it polynomially equivalent to a pair P=(A, B) if A=f(A,B), B=g(A ,B) for some polynomials f, g∈ 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 \widetildeP(\widetildeA, \widetildeB) from N(K) will be called \it polynomially similar if there exists a pair P(A, B) from N(K) such that P, P are polynomially equivalent and P, \widetildeP 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.