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Une application des nombres de Pisot à l’algorithme de Jacobi-Perron. (French) Zbl 0543.10023

Since O. Perron introduced in 1907 the Jacobi-Perron algorithm, which is the simplest generalization of continued fractions to finite sets of real numbers, the main question of characterizing the periodicity is still open. The usual conjecture is that the development of any basis of a real number field by this algorithm is periodic. But we only know some infinite families for which this is true.
In this paper we prove that for any real number field there exists a basis for which we have periodicity. For this we use the property ”The conjugates of a Pisot number are multiplicatively independent”. We also give some numerical examples.

MSC:

11J70 Continued fractions and generalizations
11A55 Continued fractions
11-04 Software, source code, etc. for problems pertaining to number theory
11R06 PV-numbers and generalizations; other special algebraic numbers; Mahler measure
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References:

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