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

Bell

doi:10.1016/j.ic.2021.104736

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

Bell

Authors

Paul Bell p.c.bell@keele.ac.uk

Abstract

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

Acceptance Date	Feb 14, 2021
Publication Date	Dec 1, 2021
Journal	Information and Computation
Print ISSN	0890-5401
Publisher	Elsevier
DOI	https://doi.org/10.1016/j.ic.2021.104736
Publisher URL	https://www.sciencedirect.com/science/article/pii/S0890540121000511?via%3Dihub