Multidimensional continued fractions (MCFs) were introduced by Jacobi and Perron in order to generalize the classical continued fractions. In this paper, we propose an introductive fundamental study about MCFs in the field of the p-adic numbers Qp. First, we introduce them from a formal point of view, i.e., without considering a specific algorithm that produces the partial quotients of an MCF, and we perform a general study about their convergence in Qp. In particular, we derive some sufficient conditions for their convergence and we prove that convergent MCFs always strongly converge in Qp contrary to the real case where strong convergence is not always guaranteed. Then, we focus on a specific algorithm that, starting from an m-tuple of numbers in Qp (p odd), produces the partial quotients of the corresponding MCF. We see that this algorithm is derived from a generalized p-adic Euclidean algorithm and we prove that it always terminates in a finite number of steps when it processes rational numbers.

On p-adic multidimensional continued fractions / Murru, Nadir; Terracini, Lea. - In: MATHEMATICS OF COMPUTATION. - ISSN 0025-5718. - 88:320(2019), pp. 2913-2934. [10.1090/mcom/3450]

### On p-adic multidimensional continued fractions

#### Abstract

Multidimensional continued fractions (MCFs) were introduced by Jacobi and Perron in order to generalize the classical continued fractions. In this paper, we propose an introductive fundamental study about MCFs in the field of the p-adic numbers Qp. First, we introduce them from a formal point of view, i.e., without considering a specific algorithm that produces the partial quotients of an MCF, and we perform a general study about their convergence in Qp. In particular, we derive some sufficient conditions for their convergence and we prove that convergent MCFs always strongly converge in Qp contrary to the real case where strong convergence is not always guaranteed. Then, we focus on a specific algorithm that, starting from an m-tuple of numbers in Qp (p odd), produces the partial quotients of the corresponding MCF. We see that this algorithm is derived from a generalized p-adic Euclidean algorithm and we prove that it always terminates in a finite number of steps when it processes rational numbers.
##### Scheda breve Scheda completa Scheda completa (DC)
2019
File in questo prodotto:
File

accesso aperto

Tipologia: 2. Post-print / Author's Accepted Manuscript
Licenza: Creative commons
Dimensione 377.7 kB
On 𝑝-adic multidimensional continued fractions.pdf

non disponibili

Tipologia: 2a Post-print versione editoriale / Version of Record
Licenza: Non Pubblico - Accesso privato/ristretto
Dimensione 266.05 kB