Quartic Salem numbers which are Mahler measures of non-reciprocal 2-Pisot numbers
Toufik Zaïmi1
1 Department of Mathematics and Statistics. College of Science Imam Mohammad Ibn Saud Islamic University (IMSIU) P. O. Box 90950 Riyadh 11623 Saudi Arabia
Journal de théorie des nombres de Bordeaux, Volume 32 (2020) no. 3, pp. 877-889.

Motivated by a question of M. J. Bertin, we obtain parametrizations of minimal polynomials of quartic Salem numbers, say α, which are Mahler measures of non-reciprocal 2-Pisot numbers. This allows us to determine all such numbers α with a given trace, and to deduce that for any natural number t (resp. t≥2) there is a quartic Salem number of trace t which is (resp. which is not) a Mahler measure of a non-reciprocal 2-Pisot number.

Motivé par une question de M. J. Bertin, on obtient des paramétrisations des polynômes minimaux des nombres de Salem quartiques, disons α, qui sont des mesures de Mahler des 2 -nombres de Pisot non-réciproques. Cela nous permet de déterminer de tels nombres α, de trace donnée, et de déduire que pour tout entier naturel t (resp. t≥2), il y a un nombre de Salem quartique, de trace t, qui est (resp. qui n’est pas) une mesure de Mahler d’un 2 -nombre de Pisot non-réciproque.

Received:
Revised:
Accepted:
Published online:
DOI: 10.5802/jtnb.1145
Classification: 11R06, 11R80, 11J71
Keywords: Salem numbers, Mahler measure, $2$-Pisot numbers.
Toufik Zaïmi 1

1 Department of Mathematics and Statistics. College of Science Imam Mohammad Ibn Saud Islamic University (IMSIU) P. O. Box 90950 Riyadh 11623 Saudi Arabia
License: CC-BY-ND 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights 
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Toufik Zaïmi. Quartic Salem numbers which are Mahler measures of non-reciprocal 2-Pisot numbers. Journal de théorie des nombres de Bordeaux, Volume 32 (2020) no. 3, pp. 877-889. doi : 10.5802/jtnb.1145. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1145/

[1] Marie José Bertin; Annette Decomps-Guilloux; Marthe Grandet-Hugot; Martine Pathiaux-Delefosse; Jean-Pierre Schreiber Pisot and Salem numbers, Birkhäuser, 1992 | Zbl

[2] Marie José Bertin; Toufik Zaïmi Complex Pisot numbers in algebraic number fields, C. R. Math. Acad. Sci. Paris, Volume 353 (2015) no. 11, pp. 965-967 | DOI | MR | Zbl

[3] David W. Boyd Inverse problems for Mahler’s measure, Diophantine analysis (London Mathematical Society Lecture Note Series), Volume 109, Cambridge University Press, 1986, pp. 147-158 | DOI | MR | Zbl

[4] David W. Boyd Reciprocal algebraic integers whose Mahler measures are non-reciprocal, Can. Math. Bull., Volume 30 (1987) no. 1-3, pp. 3-8 | DOI | Zbl

[5] David W. Boyd Salem numbers of degree four have periodic expansions, Théorie des nombres (Quebec, PQ, 1987), Walter de Gruyter, 1989, pp. 57-64 | MR | Zbl

[6] David G. Cantor On sets of algebraic integers whose remaining conjugates lie in the unit circle, Trans. Am. Math. Soc., Volume 105 (1962), pp. 391-406 | DOI | MR | Zbl

[7] Cristiane Chamfy Fonctions méromorphes dans le cercle unité et leurs séries de Taylor, Ann. Inst. Fourier, Volume 8 (1958), pp. 211-262 | DOI | Numdam | Zbl

[8] John D. Dixon; Artūras Dubickas The values of Mahler measures, Mathematika, Volume 51 (2004) no. 1-2, pp. 131-148 | DOI | MR | Zbl

[9] Artūras Dubickas On numbers which are Mahler measures, Monatsh. Math., Volume 141 (2004) no. 2, pp. 119-126 | DOI | MR | Zbl

[10] Artūras Dubickas Salem numbers as Mahler measures of nonreciprocal units, Acta Arith., Volume 176 (2016) no. 1, pp. 81-88 | MR | Zbl

[11] John B. Kelly A closed set of algebraic integers, Am. J. Math., Volume 72 (1950), pp. 565-572 | DOI | MR | Zbl

[12] Raphael Salem Power series with integral coefficients, Duke Math. J., Volume 12 (1945), pp. 153-173 | DOI | MR | Zbl

[13] Raphael Salem Algebraic numbers and Fourier analysis, Heath Mathematical Monographs, Heath and Company, 1963 | MR | Zbl

[14] The PARI Group PARI/GP version 2.5.1, 2012 (available from http://pari.math.u-bordeaux.fr/)

[15] Toufik Zaïmi On Salem numbers which are Mahler measures of non-reciprocal 2-Pisot numbers (to appear in Publ. Math. Debrecen) | Zbl

[16] Toufik Zaïmi Caractérisation d’un ensemble généralisant l’ensemble des nombres de Pisot, Acta Arith., Volume 87 (1998) no. 2, pp. 141-144 | DOI | MR | Zbl

[17] Toufik Zaïmi Sur la fermeture de l’ensemble des K-nombres de Pisot, Acta Arith., Volume 83 (1998) no. 4, pp. 363-367 | DOI | MR | Zbl

[18] Toufik Zaïmi On the distribution of powers of a Gaussian Pisot number, Indag. Math., Volume 31 (2020) no. 1, pp. 177-183 | DOI | MR | Zbl

[19] Toufik Zaïmi Salem numbers as Mahler measures of Gaussian Pisot numbers, Acta Arith., Volume 194 (2020) no. 4, pp. 383-392 | DOI | MR | Zbl

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