Bell, PC, Potapov, I and Semukhin, P (2021) On the mortality problem: From multiplicative matrix equations to linear recurrence sequences and beyond. Information and Computation, 281 (104736). ISSN 0890-5401

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We consider a variant of the mortality problem: given k × k matrices A1, . . . , At , do there exist nonnegative integers m1, . . . , mt such that Am1 1 · · · Amt t equals the zero matrix? This problem is known to be decidable when t ≤ 2 but undecidable for integer matrices with sufficiently large t and k. We prove that for t = 3 this problem is Turing-equivalent to Skolem’s problem and thus decidable for k ≤ 3 (resp. k = 4) over (resp. real) algebraic numbers. Consequently, the set of triples (m1, m2, m3) for which the equation Am1 1 Am2 2 Am3 3 equals the zero matrix is a finite union of direct products of semilinear sets. For t = 4 we show that the solution set can be non-semilinear, and thus there is unlikely to be a connection to Skolem’s problem. We prove decidability for upper-triangular 2 × 2 rational matrices by employing powerful tools from transcendence theory such as Baker’s theorem and S-unit equations

Item Type: Article
Additional Information: The final version of this article and all relevant information related to it, including copyrights, can be found on the publisher website.
Subjects: Q Science > QA Mathematics
T Technology > T Technology (General)
Divisions: Faculty of Natural Sciences > School of Computing and Mathematics
Depositing User: Symplectic
Date Deposited: 25 Jul 2022 14:18
Last Modified: 25 Jul 2022 14:18

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