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Mathematics of Computation

Published by the American Mathematical Society, the Mathematics of Computation (MCOM) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

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

The 2020 MCQ for Mathematics of Computation is 1.98.

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On the finiteness and periodicity of the $p$-adic Jacobi–Perron algorithm
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by Nadir Murru and Lea Terracini HTML | PDF
Math. Comp. 89 (2020), 2913-2930 Request permission

Abstract:

Multidimensional continued fractions (MCFs) were introduced by Jacobi and Perron to obtain periodic representations for algebraic irrationals, analogous to the case of simple continued fractions and quadratic irrationals. Continued fractions have been studied in the field of $p$-adic numbers $\mathbb {Q}_p$. MCFs have also been recently introduced in $\mathbb {Q}_p$, including, in particular, a $p$-adic Jacobi–Perron algorithm. In this paper, we address two of the main features of this algorithm, namely its finiteness and periodicity. Regarding the finiteness of the $p$-adic Jacobi–Perron algorithm, our results are obtained by exploiting properties of some auxiliary integer sequences. It is known that a finite $p$-adic MCF represents $\mathbb Q$-linearly dependent numbers. However, we see that the converse is not always true and we prove that in this case infinitely many partial quotients of the MCF have $p$-adic valuations equal to $-1$. Finally, we show that a periodic MCF of dimension $m$ converges to an algebraic irrational of degree less than or equal to $m+1$; for the case $m=2$, we are able to give some more detailed results.
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Additional Information
  • Nadir Murru
  • Affiliation: Department of Mathematics G. Peano, University of Torino, Via Carlo Alberto 10, 10123, Torino, Italy
  • MR Author ID: 905269
  • Email: nadir.murru@unito.it
  • Lea Terracini
  • Affiliation: Department of Mathematics G. Peano, University of Torino, Via Carlo Alberto 10, 10123, Torino, Italy
  • MR Author ID: 261537
  • Email: lea.terracini@unito.it
  • Received by editor(s): July 20, 2019
  • Received by editor(s) in revised form: January 21, 2020, and February 16, 2020
  • Published electronically: May 19, 2020
  • © Copyright 2020 American Mathematical Society
  • Journal: Math. Comp. 89 (2020), 2913-2930
  • MSC (2010): Primary 11J70, 12J25, 11J61
  • DOI: https://doi.org/10.1090/mcom/3540
  • MathSciNet review: 4136551