Computation of lattice isomorphisms and the integral matrix similarity problem

Abstract

AbstractLetKbe a number field, letAbe a finite-dimensionalK-algebra, let$\operatorname {\mathrm {J}}(A)$denote the Jacobson radical ofAand let$\Lambda $be an$\mathcal {O}_{K}$-order inA. Suppose that each simple component of the semisimpleK-algebra$A/{\operatorname {\mathrm {J}}(A)}$is isomorphic to a matrix ring over a field. Under this hypothesis onA, we give an algorithm that, given two$\Lambda $-latticesXandY, determines whetherXandYare isomorphic and, if so, computes an explicit isomorphism$X \rightarrow Y$. This algorithm reduces the problem to standard problems in computational algebra and algorithmic algebraic number theory in polynomial time. As an application, we give an algorithm for the following long-standing problem: Given a number fieldK, a positive integernand two matrices$A,B \in \mathrm {Mat}_{n}(\mathcal {O}_{K})$, determine whetherAandBare similar over$\mathcal {O}_{K}$, and if so, return a matrix$C \in \mathrm {GL}_{n}(\mathcal {O}_{K})$such that$B= CAC^{-1}$. We give explicit examples that show that the implementation of the latter algorithm for$\mathcal {O}_{K}=\mathbb {Z}$vastly outperforms implementations of all previous algorithms, as predicted by our complexity analysis.