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 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.