Computation of lattice isomorphisms and the integral matrix similarity problem

TL;DR

This paper presents a polynomial-time algorithm for determining lattice isomorphisms over number fields and applies it to solve the longstanding matrix similarity problem over the ring of integers, outperforming previous methods.

Contribution

The authors develop a new algorithm for lattice isomorphism over orders in semisimple algebras and apply it to efficiently solve the matrix similarity problem over number rings.

Findings

Algorithm determines lattice isomorphism in polynomial time.

Implementation outperforms previous algorithms for matrix similarity.

Explicit examples demonstrate practical efficiency and scalability.

Abstract

Let $K$ be a number field, let $A$ be a finite-dimensional $K$-algebra, let $\mathrm{J}(A)$ denote the Jacobson radical of $A$, and let $\Lambda$ be an $\mathcal{O}_{K}$-order in $A$. Suppose that each simple component of the semisimple $K$-algebra $A/{\mathrm{J}(A)}$ is isomorphic to a matrix ring over a field. Under this hypothesis on $A$, we give an algorithm that given two $\Lambda$-lattices $X$ and $Y$, determines whether $X$ and $Y$ are 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 field $K$, a positive integer $n$ and two matrices $A,B \in \mathrm{Mat}_{n}(\mathcal{O}_{K})$, determine whether $A$ and $B$ are 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.