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Mathematics of Computation

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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
Published electronically: May 17, 2019
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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.

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Additional Information

Nadir Murru
Affiliation: Department of Mathematics L. Lagrange, Politecnico of Turin, Turin, Italy

Lea Terracini
Affiliation: Department of Mathematics G. Peano, University of Turin, Turin, Italy

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