A simple, polynomial-time algorithm for the matrix torsion problem

The Matrix Torsion Problem (MTP) is: given a square matrix M with rational entries, decide whether two distinct powers of M are equal. It has been shown by Cassaigne and the author that the MTP reduces to the Matrix Power Problem (MPP) in polynomial time [1]: given two square matrices A and B with rational entries, the MTP is to decide whether B is a power of A. Since the MPP is decidable in polynomial time [3], it is also the case of the MTP. However, the algorithm for MPP is highly non-trivial. The aim of this note is to present a simple, direct, polynomialtime algorithm for the MTP.