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

##### MSC:
 11A55 Continued fractions 11B37 Recurrences 11J70 Continued fractions and generalizations
##### Keywords:
multi-continued fraction expansion; m-CFA; periodicity
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