A new multidimensional continued fraction algorithm

Abstract:

It has been believed that the continued fraction expansion of $(\alpha ,\beta )$ $(1,\alpha ,\beta$ is a ${\mathbb Q}$-basis of a real cubic field$)$ obtained by the modified Jacobi-Perron algorithm is periodic. We conducted a numerical experiment (cf. Table B, Figure 1 and Figure 2) from which we conjecture the non-periodicity of the expansion of $(\langle \sqrt [3]{3}\rangle , \langle \sqrt [3]{9}\rangle )$ ($\langle x\rangle$ denoting the fractional part of $x$). We present a new algorithm which is something like the modified Jacobi-Perron algorithm, and give some experimental results with this new algorithm. From our experiments, we can expect that the expansion of $(\alpha ,\beta )$ with our algorithm always becomes periodic for any real cubic field. We also consider real quartic fields.