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Periodic Jacobi-Perron expansions associated with a unit. (English. French summary) Zbl 1270.11068

The Jacobi-Perron algorithm (JPA) is a generalisation to higher dimension of the continued fraction algorithm. One starts with \(\alpha^{(0)} =\left(\alpha^{(0)}_1\dots , \alpha^{(0)}_n\right) \in {\mathbb R}^n\) and \(a^{(0)} =\left(a^{(0)}_1\dots , a^{(0)}_n\right) \), and by definition for \(v\geq 0\), \[ \alpha^{(v+1)} =\left(\frac{\alpha^{(v)}_2-a^{(v)}_2}{\alpha^{(v)}_1-a^{(v)}_1}, \,\dots \, , \, \frac{\alpha^{(v)}_n-a^{(v)}_n}{\alpha^{(v)}_1-a^{(v)}_1},\, \frac{1}{\alpha^{(v)}_1-a^{(v)}_1}\right) \] with \(a^{(v)}_i=\left[\alpha^{(v)}_i\right]\) for \(i=1, \dots , n\). Suppose that \(K\) is an algebraic number field of degree \(n+1\) and that the JPA of \(\alpha^{(0)} \in K^n\) has the property that \(\alpha^{(0)}=\alpha^{(\ell)}\) for some minimal integer \(\ell\). We then say that the JPA of \(\alpha^{(0)}\) is purely periodic of length \(\ell\), in which case L. Bernstein proved that \(\eta= \prod _{v=0}^{\ell -1} \alpha^{(v)}_n\) is a unit of \(K\); the authors say that this unit \(\eta \) is then associated to the vector \(\alpha^{(0)}\) of \(K^n\). In the paper under review, the authors prove that given a unit \(\varepsilon\) of degree \(n+1\) in \(K\), there are only finitely many vectors \(\alpha^{(0)} \in K^n\) whose JPA expansion is purely periodic (of length, say, \(\ell\)), and provides a Bernstein unit \(\prod _{v=1}^{\ell -1} \alpha^{(v)}_n\) which happens to be equal to that given unit \(\varepsilon\). The result is interesting and the proof is rather technical. Amazing examples are exhibited for the cubic fields

MSC:

11J70 Continued fractions and generalizations
11K50 Metric theory of continued fractions
11A55 Continued fractions
11R27 Units and factorization
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References:

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