Pisot unit generators in number fields. (English) Zbl 1398.11160
Summary: Pisot numbers are real algebraic integers bigger than 1, whose other conjugates all have modulus smaller than 1. In this paper we deal with the algorithmic problem of finding the smallest Pisot unit generating a given number field. We first solve this problem in all real fields, then we consider the analogous problem involving the so called complex Pisot numbers and we solve it in all number fields that admit such a generator, in particular this includes all fields without CM.
MSC:
| 11Y40 | Algebraic number theory computations |
| 11R06 | PV-numbers and generalizations; other special algebraic numbers; Mahler measure |
| 11R27 | Units and factorization |
Software:
SageMathReferences:
| [1] | Akiyama, S., Pisot numbers and greedy algorithm, (Number Theory. Number Theory, Eger, 1996 (1998), de Gruyter: de Gruyter Berlin), 9-21 · Zbl 0919.11063 |
| [2] | Amoroso, F.; Dvornicich, R., A lower bound for the height in abelian extensions, J. Number Theory, 80, 2, 260-272 (2000) · Zbl 0973.11092 |
| [3] | Arvind, V.; Kurur, P. P., On the complexity of computing the units in a number field, (6th Algorithmic Number Theory Symposium. 6th Algorithmic Number Theory Symposium, ANTS. 6th Algorithmic Number Theory Symposium. 6th Algorithmic Number Theory Symposium, ANTS, Lecture Notes in Computer Science, vol. 3076 (2004), Springer), 72-86 · Zbl 1148.11320 |
| [4] | Bertin, M.-J.; Decomps-Guilloux, A.; Grandet-Hugot, M.; Pathiaux-Delefosse, M.; Schreiber, J.-P., Pisot and Salem Numbers (1992), Birkhäuser Verlag: Birkhäuser Verlag Basel, with a preface by David W. Boyd · Zbl 0772.11041 |
| [5] | Cheng, Q.; Zhuang, J., On certain computations of Pisot numbers, Inf. Process. Lett., 113, 8, 271-275 (2013) · Zbl 1272.68476 |
| [6] | Cohen, H., A Course in Computational Algebraic Number Theory (1993), Springer-Verlag New York, Inc.: Springer-Verlag New York, Inc. New York, NY, USA · Zbl 0786.11071 |
| [7] | Daileda, R. C., Algebraic integers on the unit circle, J. Number Theory, 118, 2, 189-191 (2006) · Zbl 1143.11040 |
| [8] | Danna, E.; Fenelon, M.; Gu, Z.; Wunderling, R., Generating multiple solutions for mixed integer programming problems, (Integer Programming and Combinatorial Optimization. Integer Programming and Combinatorial Optimization, Lecture Notes in Comput. Sci., vol. 4513 (2007), Springer: Springer Berlin), 280-294 · Zbl 1136.90416 |
| [9] | Developers, T. S., SageMath, the Sage Mathematics Software System (Version 6.10) (2015) |
| [10] | Lenstra, J. H.W., Integer programming with a fixed number of variables, Math. Oper. Res., 8, 4, 538-548 (1983) · Zbl 0524.90067 |
| [11] | Marcus, D. A., Number Fields (1977), Springer-Verlag: Springer-Verlag New York-Heidelberg, universitext · Zbl 0383.12001 |
| [12] | Remak, R., Über algebraische Zahlkörper mit schwachem Einheitsdefekt, Compos. Math., 12, 35-80 (1954) · Zbl 0055.26805 |
| [13] | Salem, R., Power series with integral coefficients, Duke Math. J., 12, 153-172 (1945) · Zbl 0060.21601 |
| [14] | Salem, R., Algebraic Numbers and Fourier Analysis, Heath Mathematical Monographs (1963), Heath · Zbl 0126.07802 |
| [15] | Schinzel, A., On the product of the conjugates outside the unit circle of an algebraic number, Acta Arith., 24, 385-399 (1973), collection of articles dedicated to Carl Ludwig Siegel on the occasion of his seventy-fifth birthday. IV · Zbl 0275.12004 |
| [16] | Schmidt, K., On periodic expansions of Pisot numbers and Salem numbers, Bull. Lond. Math. Soc., 12, 4, 269-278 (1980) · Zbl 0494.10040 |
| [17] | Smyth, C. J., On the product of the conjugates outside the unit circle of an algebraic integer, Bull. Lond. Math. Soc., 3, 169-175 (1971) · Zbl 0235.12003 |
| [18] | Valiant, L., The complexity of computing the permanent, Theor. Comput. Sci., 8, 2, 189-201 (1979) · Zbl 0415.68008 |
| [19] | von zur Gathen, J.; Sieveking, M., A bound on solutions of linear integer equalities and inequalities, Proc. Am. Math. Soc., 72, 1, 155-158 (1978) · Zbl 0397.90071 |
| [20] | Voutier, P., An effective lower bound for the height of algebraic numbers, Acta Arith., 74, 1, 81-95 (1996) · Zbl 0838.11065 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
