## The Jacobi–Perron algorithm and Pisot numbers.(English)Zbl 1051.11037

The Jacobi-Perron algorithm is a generalization of the continued fraction algorithm. Applied to an $$n$$-uple of real numbers, it gives simultaneous approximations. In case of periodicity it yields a unit of a number field which commands the quality of simultaneous approximations.
In this paper, the authors prove that for $$n= 2$$ this unit is a Pisot number (a positive algebraic integer with each conjugate in $$| z|< 1$$) and that it is not necessary the case for $$n\geq 3$$.
Recently, B. Adam and G. Rhin found a method yielding all pairs of real numbers with periodic Jacobi-Perron algorithm which produce a given unit in a real cubic field. In many examples they get no set with periodic Jacobi-Perron algorithm when the given unit is not a Pisot number. So, they ask if this is always true. By this paper, the authors give a positive answer to their question for $$n= 2$$ and they prove that this is not always the case for $$n\geq 3$$.

### MSC:

 11J70 Continued fractions and generalizations 11R06 PV-numbers and generalizations; other special algebraic numbers; Mahler measure 11R27 Units and factorization 11R16 Cubic and quartic extensions
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