## 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

### Keywords:

Jacobi-Perron algorithms
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### References:

 [1] B. Adam, Voronoï-algorithm expansion of two families with period length going to infinity. Math. of Comp. 64 (1995), no 212, 1687-1704. · Zbl 0858.11070 [2] B. Adam and G. Rhin, Algorithme des fractions continues et de Jacobi-Perron. Bull. Austral. Math. Soc. 53 (1996), 341-350. · Zbl 0858.11069 [3] L. Bernstein, The Jacobi-Perron Algorithm, Its Theory and Application. Lecture Notes in Mathematics, 207, Springer-Verlag, Berlin-New York, 1971. · Zbl 0213.05201 [4] L. Bernstein, Einheitenberechnung in kubischen Körpern mittels des Jacobi-Perronschen Algorithmus aus der Rechenanlage. J. Reine Angew. Math. 244 (1970), 201-220. · Zbl 0205.35301 [5] L. Bernstein, A 3-Dimensional Periodic Jacobi-Perron Algorithm of Period Length 8. J. of Number Theory 4 (1972), no.1, 48-69. · Zbl 0244.10006 [6] L. Bernstein, On units and fundamental units. J. Reine Angew. Math. 257 (1972), 129-145. · Zbl 0247.12007 [7] H. Cohen, A Course in Computational Algebraic Number Theory. Graduate Texts in Maths, 138, Springer Verlag, 2007. · Zbl 0786.11071 [8] E. Dubois and R. Paysant-Le Roux, Une application des nombres de Pisot à l’algorithme de Jacobi-Perron. Monatschefte für Mathematik 98 (1984), 145-155. · Zbl 0543.10023 [9] E. Dubois, A. Farhane and R. Paysant-Le Roux, The Jacobi-Perron Algorithm and Pisot numbers. Acta Arith. 111 (2004), no 3, 269-275. · Zbl 1051.11037 [10] J. C. Lagarias, The Quality of the Diophantine Approximations found by the Jacobi-Perron Algorithm and Related Algorithms. Monatschefte für Math. 115 (1993), 299-328. · Zbl 0790.11059 [11] C. Levesque and G. Rhin, Two Families of Periodic Jacobi Algorithms with Period Lengths Going to Infinity. Jour. of Number Theory 37 (1991), no. 2, 173-180. · Zbl 0723.11032 [12] C. Batut, K. Belabas, D. Bernardi, H. Cohen and M. Olivier, GP-Pari version 2.3.4 (2009). [13] O. Perron, Grundlagen für eine Theorie des Jacobischen Kettenbruchalgorithmus. Math. Ann. 64 (1907), 1-76. · JFM 38.0262.01 [14] H. J. Stender, Eine Formel für Grundeinheiten in reinen algebraischen Zahlkörpern dritten, vierten und sechsten Grades. Jour. of Number Theory 7 (1975), no. 2, 235-250. · Zbl 0308.12001
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