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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