On the Mortality Problem: from multiplicative matrix equations to linear recurrence sequences and beyond
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We consider the following variant of the Mortality Problem: given k× k matrices A_1, A_2, ...,A_t, does there exist nonnegative integers m_1, m_2, ...,m_t such that the product A_1^m_1 A_2^m_2... A_t^m_t is equal to the zero matrix? It is known that this problem is decidable when t ≤ 2 for matrices over algebraic numbers but becomes undecidable for sufficiently large t and k even for integral matrices. In this paper, we prove the first decidability results for t>2. We show as one of our central results that for t=3 this problem in any dimension is Turing equivalent to the well-known Skolem problem for linear recurrence sequences. Our proof relies on the Primary Decomposition Theorem for matrices that was not used to show decidability results in matrix semigroups before. As a corollary we obtain that the above problem is decidable for t=3 and k ≤ 3 for matrices over algebraic numbers and for t=3 and k=4 for matrices over real algebraic numbers. Another consequence is that 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 equal to a finite union of direct products of semilinear sets. For t=4 we show that the solution set can be non-semilinear, and thus it seems unlikely that there is a direct connection to the Skolem problem. However we prove that the problem is still decidable for upper-triangular 2 × 2 rational matrices by employing powerful tools from transcendence theory such as Baker's theorem and S-unit equations.
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