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.
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.
Revised:
Accepted:
Published online:
Mots-clés : Salem numbers, Mahler measure, $2$-Pisot numbers.
@article{JTNB_2020__32_3_877_0, author = {Za{\"\i}mi, Toufik}, 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 = {http://archive.numdam.org/articles/10.5802/jtnb.1145/} }
TY - JOUR AU - Zaïmi, Toufik TI - Quartic Salem numbers which are Mahler measures of non-reciprocal 2-Pisot numbers JO - Journal de théorie des nombres de Bordeaux PY - 2020 SP - 877 EP - 889 VL - 32 IS - 3 PB - Société Arithmétique de Bordeaux UR - http://archive.numdam.org/articles/10.5802/jtnb.1145/ DO - 10.5802/jtnb.1145 LA - en ID - JTNB_2020__32_3_877_0 ER -
%0 Journal Article %A Zaïmi, Toufik %T Quartic Salem numbers which are Mahler measures of non-reciprocal 2-Pisot numbers %J Journal de théorie des nombres de Bordeaux %D 2020 %P 877-889 %V 32 %N 3 %I Société Arithmétique de Bordeaux %U http://archive.numdam.org/articles/10.5802/jtnb.1145/ %R 10.5802/jtnb.1145 %G en %F JTNB_2020__32_3_877_0
Zaïmi, Toufik. 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. http://archive.numdam.org/articles/10.5802/jtnb.1145/
[1] Pisot and Salem numbers, Birkhäuser, 1992 | Zbl
[2] 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] 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] Reciprocal algebraic integers whose Mahler measures are non-reciprocal, Can. Math. Bull., Volume 30 (1987) no. 1-3, pp. 3-8 | DOI | Zbl
[5] Salem numbers of degree four have periodic expansions, Théorie des nombres (Quebec, PQ, 1987), Walter de Gruyter, 1989, pp. 57-64 | MR | Zbl
[6] 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] Fonctions méromorphes dans le cercle unité et leurs séries de Taylor, Ann. Inst. Fourier, Volume 8 (1958), pp. 211-262 | DOI | Numdam | Zbl
[8] The values of Mahler measures, Mathematika, Volume 51 (2004) no. 1-2, pp. 131-148 | DOI | MR | Zbl
[9] On numbers which are Mahler measures, Monatsh. Math., Volume 141 (2004) no. 2, pp. 119-126 | DOI | MR | Zbl
[10] Salem numbers as Mahler measures of nonreciprocal units, Acta Arith., Volume 176 (2016) no. 1, pp. 81-88 | MR | Zbl
[11] A closed set of algebraic integers, Am. J. Math., Volume 72 (1950), pp. 565-572 | DOI | MR | Zbl
[12] Power series with integral coefficients, Duke Math. J., Volume 12 (1945), pp. 153-173 | DOI | MR | Zbl
[13] Algebraic numbers and Fourier analysis, Heath Mathematical Monographs, Heath and Company, 1963 | MR | Zbl
[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) | Zbl
[16] 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] Sur la fermeture de l’ensemble des -nombres de Pisot, Acta Arith., Volume 83 (1998) no. 4, pp. 363-367 | DOI | MR | Zbl
[18] On the distribution of powers of a Gaussian Pisot number, Indag. Math., Volume 31 (2020) no. 1, pp. 177-183 | DOI | MR | Zbl
[19] Salem numbers as Mahler measures of Gaussian Pisot numbers, Acta Arith., Volume 194 (2020) no. 4, pp. 383-392 | DOI | MR | Zbl
Cited by Sources: