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.

中文翻译：

---

关于$p$-adic Jacobi-Perron算法的有限性和周期性

Jacobi 和 Perron 引入了多维连分数 (MCF)，以便获得代数无理数的周期性表示，就像连分数和二次无理数一样。由于在 $p$--adic 数 $\mathbb Q_p$ 领域也研究了连分数，最近在 $\mathbb Q_p$ 中也引入了 MCFs 到 $p$--adic Jacobi--Perron 算法. 在本文中，我们研究了该算法的两个主要特征，即有限性和周期性。特别是，关于 $p$--adic Jacobi--Perron 算法的有限性，我们的结果是通过利用一些辅助整数序列的性质获得的。此外，已知有限的$p$--adic MCF 表示$\mathbb Q$--线性相关数。我们看到反之亦然并不总是正确的，我们证明在这种情况下，MCF 的无限部分商具有 $p$--adic 估值等于 $-1$。最后，我们证明了维度 $m$ 的周期性 MCF 收敛到次数小于或等于 $m+1$ 的代数无理数，并且对于 $m=2$ 的情况，我们能够给出一些更详细的结果。