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Decidability of multiplicative matrix equations and related Diophantine problems

Sebastian Heintze, Armand Noubissie and Robert F. Tichy
Vol. 14 (2025), No. 3-4, 251–270
DOI: 10.2140/cnt.2025.14.251
Abstract

Some new decidability results for multiplicative matrix equations over algebraic number fields are established. In particular, special instances of the so-called knapsack problem are considered. The proofs are based on effective methods for Diophantine problems in finitely generated domains. In particular, we apply results from Győry (2019) on S-unit equations and a version of a p-adic subspace theorem due to Corvaja and Zannier (2002). The focus lies on explicit bounds for the size of the solutions in terms of heights as well as on bounds for the number of solutions. This approach also works for systems of symmetric matrices which do not form a semigroup. In the final section some related counting problems are investigated.

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Keywords
Diophantine problems, matrix equations, decidability
Mathematical Subject Classification
Primary: 11C20, 11D61
Milestones
Received: 4 June 2025
Revised: 11 September 2025
Accepted: 25 September 2025
Published: 18 November 2025
Authors
Sebastian Heintze
Institute for Analysis and Number Theory
Graz University of Technology
Graz
Austria		 FB Artificial Intelligence and Human Interfaces
University of Salzburg
Salzburg
Austria
Armand Noubissie
Institute for Analysis and Number Theory
Graz University of Technology
Graz
Austria
Robert F. Tichy
Institute for Analysis and Number Theory
Graz University of Technology
Graz
Austria
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