Computation of lattice isomorphisms and the integral matrix similarity problem

Let K be a number field, let A be a finite-dimensional K-algebra, let $\operatorname {\mathrm {J}}(A)$ 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 $A/{\operatorname {\mathrm {J}}(A)}$ is isom...

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Veröffentlicht in:Forum of Mathematics, Sigma Jg. 10
Hauptverfasser: Bley, Werner, Hofmann, Tommy, Johnston, Henri
Format: Journal Article
Sprache:Englisch
Veröffentlicht: Cambridge, UK Cambridge University Press 01.01.2022
Schlagworte:
Algebra
Algorithms
Decomposition
Fields (mathematics)
Hypotheses
Isomorphism
Lattices (mathematics)
Matrix algebra
Number theory
Polynomials
11R33
20G30
16Z05
11Y40
ISSN:2050-5094, 2050-5094
Online-Zugang:Volltext
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Zusammenfassung:Let K be a number field, let A be a finite-dimensional K-algebra, let $\operatorname {\mathrm {J}}(A)$ 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 $A/{\operatorname {\mathrm {J}}(A)}$ 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.
Bibliographie:ObjectType-Article-1
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ISSN:2050-5094
2050-5094
DOI:10.1017/fms.2022.74