On the mortality problem: From multiplicative matrix equations to linear recurrence sequences and beyond

• Published in: Information and computation Vol. 281; p. 104736
• Main Authors: Bell, Paul C., Potapov, Igor, Semukhin, Pavel
• Format: Journal Article
• Language: English
• Published: Elsevier Inc 01-12-2021
• Subjects: Baker's theorem; Linear recurrence sequences; Matrix equations; Mortality problem; Primary decomposition theorem; Skolem's problem; Skolem's problem; Matrix equations; Linear recurrence sequences; Mortality problem; Primary decomposition theorem; Baker's theorem
• ISSN: 0890-5401; 1090-2651
• DOI: 10.1016/j.ic.2021.104736

We consider a variant of the mortality problem: given k×k matrices A1,…,At, do there exist nonnegative integers m1,…,mt such that A1m1⋯Atmt 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 A1m1A2m2A3m3 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.