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On the finiteness and periodicity of the pp--adic Jacobi--Perron algorithm

Abstract

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 pp--adic numbers Qp\mathbb Q_p, also MCFs have been recently introduced in Qp\mathbb Q_p together to a pp--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 pp--adic Jacobi--Perron algorithm our results are obtained by exploiting properties of some auxiliary integer sequences. Moreover, it is known that a finite pp--adic MCF represents Q\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 pp--adic valuations equal to 1-1. Finally, we show that a periodic MCF of dimension mm converges to algebraic irrationals of degree less or equal than m+1m+1 and for the case m=2m=2 we are able to give some more detailed results

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