# On the finiteness and periodicity of the $p$--adic Jacobi--Perron algorithm

## TL;DR

This paper investigates the properties of the $p$-adic Jacobi--Perron algorithm, focusing on conditions for finiteness and periodicity, and their implications for representing algebraic irrationals and related number dependencies.

## Contribution

It provides new insights into the finiteness and periodicity of the $p$-adic Jacobi--Perron algorithm, including conditions for convergence and representation of algebraic irrationals.

## Key findings

- Finite $p$-adic MCFs represent $ ext{Q}$-linearly dependent numbers.
- Infinite partial quotients with $p$-adic valuation -1 occur in certain cases.
- Periodic MCFs of dimension $m$ converge to algebraic irrationals of degree ≤ $m+1$.

## Abstract

Multidimensional continued fractions (MCFs) were introduced by Jacobi and Perron in order to obtain periodic representations for algebraic irrationals, as it is for continued fractions and quadratic irrationals. Since continued fractions have been also studied in the field of $p$--adic numbers $\mathbb Q_p$, also MCFs have been recently introduced in $\mathbb Q_p$ together to a $p$--adic Jacobi--Perron algorithm. In this paper, we address th study of two main features of this algorithm, i.e., finiteness and periodicity. In particular, regarding the finiteness of the $p$--adic Jacobi--Perron algorithm our results are obtained by exploiting properties of some auxiliary integer sequences. Moreover, it is known that a finite $p$--adic MCF represents $\mathbb Q$--linearly dependent numbers. We see that the viceversa is not always true and we prove that in this case infinite partial quotients of the MCF have $p$--adic valuations equal to $-1$. Finally, we show that a periodic MCF of dimension $m$ converges to algebraic irrationals of degree less or equal than $m+1$ and for the case $m=2$ we are able to give some more detailed results.