Computation of lattice isomorphisms and the integral matrix similarity problem

Abstract

Let K be a number field, let A be a finite-dimensional K-algebra, let J(A) denote the Jacobson radical of A, and let Λ be an OK-order in A. Suppose that each simple component of the semisimple K-algebra A/J(A) is isomorphic to a matrix ring over a field. Under this hypothesis on A, we give an algorithm that given two Λ-lattices X and Y , determines whether X and Y are isomorphic, and if so, computes an explicit isomorphism X → Y . This algorithm reduces the problem to standard problems in computational algebra and algorithmic algebraic number theory in polynomial time. As an application, we give an algorithm for the following long-standing problem: given a number field K, a positive integer n and two matrices A, B ∈ Matn(OK), determine whether A and B are similar over OK, and if so, return a matrix C ∈ GLn(OK) such that B = CAC−1. We give explicit examples that show that the implementation of the latter algorithm for OK = Z vastly outperforms implementations of all previous algorithms, as predicted by our complexity analysis.