Study of interruptions in the Jacobi-Perron algorithm

It is well known that in general the Jacobi-Perron algorithm (a multi-dimensional analogue of the continued fraction algorithm) may or might not acknowledge the dependence over Q of its arguments 1, alpha(1),..., alpha(n) by truncating itself down to fewer arguments from some step onwards (if so, the algorithm is said to display an 'interruption'). We show here that if n = 2 then 1, alpha(1), alpha(2) are linearly dependent over Q if and only if the Jacobi-Perron Algorithm displays an interruption. We give examples showing this is not so for any n greater than or equal to 3.