A new multidimensional continued fraction algorithm

Abstract

It has been believed that the continued fraction expansion of(α,β)(\alpha ,\beta )(1,α,β(1,\alpha ,\betais aQ{\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(⟨33⟩,⟨93⟩)(\langle \sqrt [3]{3}\rangle , \langle \sqrt [3]{9}\rangle )(⟨x⟩\langle x\rangledenoting the fractional part ofxx). 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(α,β)(\alpha ,\beta )with our algorithm always becomes periodic for any real cubic field. We also consider real quartic fields.