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Eventual periodicity of the Smith forms of integer matrix powers

Vanni Noferini

TL;DR

This paper proves that the Smith form of integer matrix powers is eventually periodic up to a fixed diagonal multiplier: there exist n0, T and a diagonal D with $\mathrm{SF}(A^{n+1})=D\mathrm{SF}(A^n)$ for all large n. It shows that a stronger exact periodicity cannot hold in general, providing counterexamples where n0 and T must be large; the authors develop a p-adic framework to analyze valuations and determinantal data, and prove a linear-plus-eventually-periodic form for $\nu_p(A^n)$ via Jordan decomposition. The core method links determinantal divisors to gcd growth, uses C_r identities, and reduces the problem to p-adic valuation behavior of eigenvalues and Jordan blocks. The results clarify the asymptotic structure of invariant factors of matrix powers and have implications for areas like algebraic topology and K-theory.

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 $M \in \Z^{m \times m}$, then for every $A \in \Z^{m \times m}$ there exist $n_0 \in \N$, an integer $T \geq 1$, and a constant diagonal matrix $D \in \Z^{m \times m}$ 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.

Eventual periodicity of the Smith forms of integer matrix powers

TL;DR

This paper proves that the Smith form of integer matrix powers is eventually periodic up to a fixed diagonal multiplier: there exist n0, T and a diagonal D with for all large n. It shows that a stronger exact periodicity cannot hold in general, providing counterexamples where n0 and T must be large; the authors develop a p-adic framework to analyze valuations and determinantal data, and prove a linear-plus-eventually-periodic form for via Jordan decomposition. The core method links determinantal divisors to gcd growth, uses C_r identities, and reduces the problem to p-adic valuation behavior of eigenvalues and Jordan blocks. The results clarify the asymptotic structure of invariant factors of matrix powers and have implications for areas like algebraic topology and K-theory.

Abstract

We prove that the Smith forms of the powers of an integer square matrix behave in an eventually periodic manner. More precisely, if denotes the Smith form of , then for every there exist , an integer , and a constant diagonal matrix such that implies . This provides an eventually affirmative answer to a conjecture posed in 2013 by R. Bruner. We also show that both and can be arbitrarily large.

Paper Structure

This paper contains 8 sections, 13 theorems, 29 equations.

Table of Contents

  1. Introduction
  2. Background
  3. Periodic sequences
  4. Matrix theory
  5. Proof of Proposition \ref{['prop:no']}
  6. $p$-adic analysis
  7. Proof of Theorem \ref{['thm3']}
  8. Acknowledgements

Key Result

Theorem 1.1

Let $A \in \mathbb{Z}^{m \times m}$ and define the sequence $(S_n)_{n \in \mathbb{N}}$, where $S_n$ is the Smith form of $A^n$ normalized to have nonnegative diagonal entries. Then, there exists an eventually periodic sequence $(D_n)_{n \in \mathbb{N}} \subset \mathbb{Z}^{m \times m}$ of diagonal ma

Theorems & Definitions (23)

  • Theorem 1.1
  • Proposition 1.2
  • Proposition 2.1
  • proof
  • Proposition 2.2
  • proof
  • Theorem 2.3: Smith's Theorem, kaplanskysmith
  • Theorem 2.4
  • Theorem 2.5
  • proof : Proof of Proposition \ref{['prop:no']}
  • ...and 13 more