A criterion for periodicity of multi-continued fraction expansion of multi-formal Laurent series. (English) Zbl 1180.11003

Dai, Zongduo; Wang, Ping; Wang, Kunpeng; Feng, Xiutao

Acta Arith. 130, No. 2, 127-140 (2007).

The multi-dimensional continued fraction algorithm (m-CFA for short) over the formal Laurent series field \(F((z^{-1}))\), as a generalization of the classical continued fraction algorithm, was presented in [Z. Dai, K. Wang, D. Ye, m-continued fraction expansions of multi-Laurent series, Adv. Math., Beijing 33, 246–248 (2004) and Acta Arith. 122, 1–16 (2006; Zbl 1146.11036)]. The output of m-CFA with input \((\underline{r})\) is called the multi-continued fraction expansion of \((\underline{r})\). The classical continued fraction expansion \([a_0; a_1, a_2, \dots]\) is called \((\lambda,T)\)-periodic if there exist integers \(\lambda\geq 1\) and \(T\geq 1\) such that \(a_{\lambda+T+k}=a_{\lambda+k}\) for all \(k\geq 0\).
In this paper, the authors provide a criterion to determine whether a multi-continued fraction expansion is \((\lambda,T)\)-periodic only by means of the data obtained in the process of the m-CFA.

Reviewer: Takao Komatsu (Hirosaki)

MSC:

11A55	Continued fractions
11B37	Recurrences
11J70	Continued fractions and generalizations

Keywords:

multi-continued fraction expansion; m-CFA; periodicity