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


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