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 $\mathbb Q_p$. First, we introduce them from a formal point of view, i.e., without considering a specific algorithm that produces the partial quotients of a MCF, and we perform a general study about their convergence in $\mathbb Q_p$. In particular, we derive some conditions about their convergence and we prove that convergent MCFs always strongly converge in $\mathbb Q_p$ contrarily to the real case where strong convergence is not ever guaranteed. Then, we focus on a specific algorithm that, starting from a $m$--tuple of numbers in $\mathbb Q_p$, 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.