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. ISSN 08905401

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Abstract
We consider a variant of the mortality problem: given matrices $A_1, ..., A_t$, do there exist nonnegative integers $m_1, \ldots, m_t$ such that $A_1^{m_1} \cdots A_t^{m_t}$ equals the zero matrix? This problem is known to be decidable when $t \leq 2$ but undecidable for integer matrices with sufficiently large $t$ and $k$. We prove that for t=3 this problem is Turingequivalent to Skolem's problem and thus decidable for k <= 3 (resp. k=4) over (resp. real) algebraic numbers. Consequently, the set of triples $(m_1, m_2, m_3)$ for which the equation $A_1^{m_1}A_2^{m_2}A_3^{m_3}$ 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 nonsemilinear, and thus there is unlikely to be a connection to Skolem's problem. We prove decidability for uppertriangular 2x2 rational matrices by employing powerful tools from transcendence theory such as Baker's theorem and Sunit equations.
Item Type:  Article 

Uncontrolled Keywords:  08 Information and Computing Sciences 
Subjects:  Q Science > QA Mathematics Q Science > QA Mathematics > QA75 Electronic computers. Computer science Q Science > QA Mathematics > QA76 Computer software 
Divisions:  Computer Science & Mathematics 
Publisher:  Elsevier 
Date Deposited:  01 Jul 2021 08:55 
Last Modified:  18 Feb 2022 00:50 
DOI or ID number:  10.1016/j.ic.2021.104736 
URI:  https://researchonline.ljmu.ac.uk/id/eprint/15200 
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