Abstract:We prove that the Smith forms of the powers of an integer square matrix behave in an eventually periodic manner. More precisely, if $\mathrm{SF}(M)$ denotes the Smith form of $M \in \mathbb{Z}^{m \times m}$, then for every $A \in \mathbb{Z}^{m \times m}$ there exist $n_0 \in \mathbb{N}$, an integer $T \geq 1$, and a constant diagonal matrix $D \in \mathbb{Z}^{m \times m}$ such that $n \geq n_0$ implies $\mathrm{SF}(A^{n+T})=D \cdot \mathrm{SF}(A^n)$. This provides an eventually affirmative answer to a conjecture posed in 2013 by R. Bruner. We also show that both $n_0$ and $T$ can be arbitrarily large.
From: Vanni Noferini [view email]
[v1]
Fri, 28 Nov 2025 00:12:26 UTC (12 KB)
[v2]
Thu, 11 Jun 2026 20:38:41 UTC (12 KB)
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