Decidability of multiplicative matrix equations and related Diophantine problems

TL;DR

This paper establishes new decidability results for multiplicative matrix equations over algebraic number fields, focusing on explicit bounds for solutions and extending to non-semigroup matrix systems and related counting problems.

Contribution

It introduces novel decidability results for matrix equations and Diophantine problems, with explicit bounds and applicability to non-semigroup matrix systems.

Findings

Decidability results for specific matrix equations over algebraic number fields.

Explicit bounds on solution sizes and counts.

Extension to systems of symmetric matrices not forming a semigroup.

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 as presented in the recent book of Evertse and Gy\"ory. 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.