1. On the mortality problem: From multiplicative matrix equations to linear recurrence sequences and beyond.
- Author
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Bell, Paul C., Potapov, Igor, and Semukhin, Pavel
- Subjects
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LINEAR equations , *ALGEBRAIC numbers , *MATRICES (Mathematics) , *MORTALITY , *INTEGERS - Abstract
We consider a variant of the mortality problem: given k × k matrices A 1 , ... , A t , do there exist nonnegative integers m 1 , ... , m t such that A 1 m 1 ⋯ A t m 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 (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 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. [ABSTRACT FROM AUTHOR]
- Published
- 2021
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