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Ducci Matrices in $p$-adic Context

Piero Giacomelli

Abstract

In this paper, we mutuate the concept of Ducci matrices to the $p$-adic setting, generalizing the classical Ducci sequences to the framework of $p$-adic numbers. The classical Ducci operator, which iteratively computes the absolute differences of neighboring elements in a sequence or matrix, is redefined using the $p$-adic absolute value $| \cdot |_p$. We investigate the dynamics of $p$-adic Ducci sequences for matrices over $\mathbb{Q}_p$, focusing on their convergence and periodicity properties.

Ducci Matrices in $p$-adic Context

Abstract

In this paper, we mutuate the concept of Ducci matrices to the -adic setting, generalizing the classical Ducci sequences to the framework of -adic numbers. The classical Ducci operator, which iteratively computes the absolute differences of neighboring elements in a sequence or matrix, is redefined using the -adic absolute value . We investigate the dynamics of -adic Ducci sequences for matrices over , focusing on their convergence and periodicity properties.

Paper Structure

This paper contains 5 theorems, 58 equations.

Key Result

Proposition 2

If all eigenvalues of $D_p \in \mathbb{M}_{n\times n} (\mathbb{Q}_p)$ have $p$-adic norm less that $1$ then the $p$-adic Ducci sequence terminates.

Theorems & Definitions (15)

  • Definition 1
  • Definition 2
  • Definition 3
  • Definition 4: $p$-adic Ducci sequence
  • Remark 1
  • Proposition 2
  • proof
  • Corollary 3
  • Theorem 4
  • proof
  • ...and 5 more