DeepAI

# On the Mortality Problem: from multiplicative matrix equations to linear recurrence sequences and beyond

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.

02/26/2019

### Decidability of the Mortality Problem: from multiplicative matrix equations to linear recurrence sequences and beyond

We consider the following variant of the Mortality Problem: given k× k m...
03/09/2021

### Improved upper bounds for the rigidity of Kronecker products

The rigidity of a matrix A for target rank r is the minimum number of en...
12/19/2019

### Analysis of the multiplicative Schwarz method for matrices with a special block structure

We analyze the convergence of the (algebraic) multiplicative Schwarz met...
06/24/2021

### Slack matrices, k-products, and 2-level polytopes

In this paper, we study algorithmic questions concerning products of mat...
01/30/2019

### The Semialgebraic Orbit Problem

The Semialgebraic Orbit Problem is a fundamental reachability question t...
07/20/2021

### Cosine and Computation

We are interested in solving decision problem ∃? t ∈ℕ, cos t θ = c where...
08/11/2019

### Bijective recurrences concerning two Schröder triangles

Let r(n,k) (resp. s(n,k)) be the number of Schröder paths (resp. little ...