Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)

 
 

 

On $ p$-adic multidimensional continued fractions


Authors: Nadir Murru and Lea Terracini
Journal: Math. Comp. 88 (2019), 2913-2934
MSC (2010): Primary 11J70, 12J25, 11J61
DOI: https://doi.org/10.1090/mcom/3450
Published electronically: May 17, 2019
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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 $ \mathbb{Q}_p$. 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 $ \mathbb{Q}_p$. In particular, we derive some sufficient conditions for their convergence and we prove that convergent MCFs always strongly converge in $ \mathbb{Q}_p$ 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 $ \mathbb{Q}_p$ ($ 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.


References [Enhancements On Off] (What's this?)


Similar Articles

Retrieve articles in Mathematics of Computation with MSC (2010): 11J70, 12J25, 11J61

Retrieve articles in all journals with MSC (2010): 11J70, 12J25, 11J61


Additional Information

Nadir Murru
Affiliation: Department of Mathematics L. Lagrange, Politecnico of Turin, Turin, Italy
Email: nadir.murru@gmail.com

Lea Terracini
Affiliation: Department of Mathematics G. Peano, University of Turin, Turin, Italy
Email: lea.terracini@unito.it

DOI: https://doi.org/10.1090/mcom/3450
Keywords: Continued fractions, Jacobi--Perron algorithm, multidimensional continued fractions, p-adic numbers
Received by editor(s): April 20, 2018
Received by editor(s) in revised form: October 31, 2018, and January 21, 2019
Published electronically: May 17, 2019
Article copyright: © Copyright 2019 American Mathematical Society