Motivé par une question de M. J. Bertin, on obtient des paramétrisations des polynômes minimaux des nombres de Salem quartiques, disons , qui sont des mesures de Mahler des -nombres de Pisot non-réciproques. Cela nous permet de déterminer de tels nombres , de trace donnée, et de déduire que pour tout entier naturel (resp. , il y a un nombre de Salem quartique, de trace , qui est (resp. qui n’est pas) une mesure de Mahler d’un -nombre de Pisot non-réciproque.
Motivated by a question of M. J. Bertin, we obtain parametrizations of minimal polynomials of quartic Salem numbers, say , which are Mahler measures of non-reciprocal -Pisot numbers. This allows us to determine all such numbers with a given trace, and to deduce that for any natural number (resp. there is a quartic Salem number of trace which is (resp. which is not) a Mahler measure of a non-reciprocal -Pisot number.
Révisé le : 2020-09-24
Accepté le : 2020-10-24
Publié le : 2021-01-08
Classification : 11R06, 11R80, 11J71
Mots clés : Salem numbers, Mahler measure, -Pisot numbers.
@article{JTNB_2020__32_3_877_0, author = {Toufik Za\"\i mi}, title = {Quartic Salem numbers which are Mahler measures of non-reciprocal 2-Pisot numbers}, journal = {Journal de Th\'eorie des Nombres de Bordeaux}, pages = {877--889}, publisher = {Soci\'et\'e Arithm\'etique de Bordeaux}, volume = {32}, number = {3}, year = {2020}, doi = {10.5802/jtnb.1145}, language = {en}, url = {https://jtnb.centre-mersenne.org/item/JTNB_2020__32_3_877_0/} }
Toufik Zaïmi. Quartic Salem numbers which are Mahler measures of non-reciprocal 2-Pisot numbers. Journal de Théorie des Nombres de Bordeaux, Tome 32 (2020) no. 3, pp. 877-889. doi : 10.5802/jtnb.1145. https://jtnb.centre-mersenne.org/item/JTNB_2020__32_3_877_0/
[1] Pisot and Salem numbers, Birkhäuser, 1992
[2] Complex Pisot numbers in algebraic number fields, C. R. Math. Acad. Sci. Paris, Volume 353 (2015) no. 11, pp. 965-967 | Zbl 1332.11093
[3] Inverse problems for Mahler’s measure, Diophantine analysis (London Mathematical Society Lecture Note Series) Volume 109, Cambridge University Press, 1986, pp. 147-158
[4] Reciprocal algebraic integers whose Mahler measures are non-reciprocal, Can. Math. Bull., Volume 30 (1987) no. 1-3, pp. 3-8
[5] Salem numbers of degree four have periodic expansions, Théorie des nombres (Quebec, PQ, 1987), Walter de Gruyter, 1989, pp. 57-64
[6] On sets of algebraic integers whose remaining conjugates lie in the unit circle, Trans. Am. Math. Soc., Volume 105 (1962), pp. 391-406 | Zbl 01235.09002
[7] Fonctions méromorphes dans le cercle unité et leurs séries de Taylor, Ann. Inst. Fourier, Volume 8 (1958), pp. 211-262 | Zbl 0087.07603
[8] The values of Mahler measures, Mathematika, Volume 51 (2004) no. 1-2, pp. 131-148
[9] On numbers which are Mahler measures, Monatsh. Math., Volume 141 (2004) no. 2, pp. 119-126 | Zbl 1065.11084
[10] Salem numbers as Mahler measures of nonreciprocal units, Acta Arith., Volume 176 (2016) no. 1, pp. 81-88
[11] A closed set of algebraic integers, Am. J. Math., Volume 72 (1950), pp. 565-572
[12] Power series with integral coefficients, Duke Math. J., Volume 12 (1945), pp. 153-173
[13] Algebraic numbers and Fourier analysis, Heath Mathematical Monographs, Heath and Company, 1963
[14] PARI/GP version 2.5.1, 2012 (available from http://pari.math.u-bordeaux.fr/)
[15] On Salem numbers which are Mahler measures of non-reciprocal -Pisot numbers (to appear in Publ. Math. Debrecen)
[16] Caractérisation d’un ensemble généralisant l’ensemble des nombres de Pisot, Acta Arith., Volume 87 (1998) no. 2, pp. 141-144
[17] Sur la fermeture de l’ensemble des -nombres de Pisot, Acta Arith., Volume 83 (1998) no. 4, pp. 363-367
[18] On the distribution of powers of a Gaussian Pisot number, Indag. Math., Volume 31 (2020) no. 1, pp. 177-183
[19] Salem numbers as Mahler measures of Gaussian Pisot numbers, Acta Arith., Volume 194 (2020) no. 4, pp. 383-392