Linear relations with conjugates of a Salem number
Journal de Théorie des Nombres de Bordeaux, Tome 32 (2020) no. 1, pp. 179-191.

Dans cet article, nous considérons les relations linéaires entre les conjugués d’un nombre de Salem α. Nous montrons qu’une telle relation provient d’une relation linéaire entre les conjugués de l’entier algébrique totalement réel correspondant α+1/α. On montre également que le plus petit degré d’un nombre de Salem satisfaisant à une relation non triviale entre ces conjugués est 8 tandis que la longueur la plus courte d’une relation linéaire non-triviale entre les conjugués d’un nombre de Salem est 6.

In this paper we consider linear relations with conjugates of a Salem number α. We show that every such a relation arises from a linear relation between conjugates of the corresponding totally real algebraic integer α+1/α. It is also shown that the smallest degree of a Salem number with a nontrivial relation between its conjugates is 8, whereas the smallest length of a nontrivial linear relation between the conjugates of a Salem number is 6.

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DOI : https://doi.org/10.5802/jtnb.1116
Classification : 11R06,  11R09
Mots clés : Les relations linéaires additives, les nombres de Salem, les nombres de Pisot, les nombres algébriques totalement réels
@article{JTNB_2020__32_1_179_0,
     author = {Art\={u}ras Dubickas and Jonas Jankauskas},
     title = {Linear relations with conjugates of a {Salem} number},
     journal = {Journal de Th\'eorie des Nombres de Bordeaux},
     pages = {179--191},
     publisher = {Soci\'et\'e Arithm\'etique de Bordeaux},
     volume = {32},
     number = {1},
     year = {2020},
     doi = {10.5802/jtnb.1116},
     language = {en},
     url = {https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1116/}
}
Artūras Dubickas; Jonas Jankauskas. Linear relations with conjugates of a Salem number. Journal de Théorie des Nombres de Bordeaux, Tome 32 (2020) no. 1, pp. 179-191. doi : 10.5802/jtnb.1116. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1116/

[1] Gerd Baron; Michael Drmota; Mariusz Skałba Polynomial relations between polynomial roots, J. Algebra, Volume 177 (1995), pp. 827-846 | Article | MR 1358487 | Zbl 0835.12003

[2] Frits Beukers; Christopher J. Smyth Cyclotomic points on curves, Number theory for the millennium I. Proceedings of the millennial conference on number theory, Urbana-Champaign, IL, USA, May 21–26, 2000, A K Peters, 2002, pp. 21-26 | Zbl 1029.11009

[3] David W. Boyd Small Salem numbers, Duke Math. J., Volume 44 (1977), pp. 315-328 | Article | MR 453692 | Zbl 0353.12003

[4] Christos Christopoulos; James McKee Galois theory of Salem polynomials, Proc. Camb. Philos. Soc., Volume 148 (2010) no. 1, pp. 47-54 | Article | MR 2575371 | Zbl 1253.12001

[5] John D. Dixon Polynomials with nontrivial relations between their roots, Acta Arith., Volume 82 (1997) no. 3, pp. 293-302 | Article | MR 1482892 | Zbl 0881.12001

[6] Michael Drmota; Mariusz Skałba On multiplicative and linear independence of polynomial roots, Proceedings of the Vienna conference, June 14-17, 1990, Vienna, Austria (Contributions to General Algebra) Volume 7, Hölder-Pichler-Tempsky; Teubner, 1991, pp. 127-135 | Zbl 0757.12001

[7] Michael Drmota; Mariusz Skałba Relations between polynomial roots, Acta Arith., Volume 71 (1995) no. 1, pp. 65-77 | Article | MR 1338672 | Zbl 0818.11038

[8] Artūras Dubickas On the degree of a linear form in conjugates of an algebraic number, Ill. J. Math., Volume 46 (2002) no. 2, pp. 571-585 | Article | MR 1936938 | Zbl 1028.11066

[9] Artūras Dubickas Additive relations with conjugate algebraic numbers, Acta Arith., Volume 107 (2003) no. 1, pp. 35-43 | Article | MR 1956984 | Zbl 1158.11341

[10] Artūras Dubickas; Kevin G. Hare; Jonas Jankauskas No two non-real conjugates of a Pisot number have the same imaginary part, Math. Comput., Volume 86 (2017) no. 304, pp. 935-950 | Article | MR 3584555 | Zbl 1361.11062

[11] Artūras Dubickas; Jonas Jankauskas Simple linear relations between conjugate algebraic numbers of low degree, J. Ramanujan Math. Soc., Volume 30 (2015) no. 2, pp. 219-235 | MR 3357522 | Zbl 1425.11169

[12] Artūras Dubickas; Christopher J. Smyth On the lines passing through two conjugates of a Salem number, Proc. Camb. Philos. Soc., Volume 144 (2008) no. 1, pp. 29-37 | Article | MR 2388230 | Zbl 1166.11031

[13] Michael Filaseta; Kevin Ford; Sergeĭ Konyagin On an irreducibility theorem of A. Schinzel associated with coverings of the integers, Ill. J. Math., Volume 44 (2000) no. 3, pp. 633-643 | Article | MR 1772434 | Zbl 0966.11046

[14] Michael Filaseta; Manton jun. Matthews On the irreducibility of 0, 1-polynomials of the form f(x)x n +g(x), Colloq. Math., Volume 99 (2004) no. 1, pp. 1-5 | Article | MR 2084532 | Zbl 1060.11066

[15] Kurt Girstmair Linear relations between roots of polynomials, Acta Arith., Volume 89 (1999) no. 1, pp. 53-96 (corrigendum in ibid. 110 (2003), no. 2, p. 203) | Article | MR 1692195 | Zbl 0924.12002

[16] V. A. Kurbatov On equations of prime degree, Mat. Sb., N. Ser., Volume 43 (1957), pp. 349-366 | MR 95159 | Zbl 0080.01404

[17] V. A. Kurbatov Linear dependence of conjugate elements, Mat. Sb., N. Ser., Volume 52 (1960), pp. 701-708 | MR 122813 | Zbl 0111.01901

[18] V. A. Kurbatov Galois extensions of prime degree and their primitive elements, Izv. Vyssh. Uchebn. Zaved., Mat., Volume 21 (1977), pp. 49-52 | MR 444624 | Zbl 0356.12027

[19] Franck Lalande La relation linéaire a=b+c++t entre les racines d’un polynôme, J. Théor. Nombres Bordeaux, Volume 19 (2007) no. 2, pp. 473-484 | Article | Numdam | MR 2394897 | Zbl 1200.11083

[20] Franck Lalande À propos de la relation galoisienne x 1 =x 2 +x 3 , J. Théor. Nombres Bordeaux, Volume 22 (2010) no. 3, pp. 661-673 | MR 2769337 | Zbl 1254.12007

[21] James McKee; Christopher J. Smyth There are Salem numbers of every trace, Bull. Lond. Math. Soc., Volume 37 (2005) no. 1, pp. 25-36 | Article | MR 2105815 | Zbl 1166.11349

[22] Raphael Salem A Remarkable Class of Algebraic Numbers. Proof of a Conjecture of Vijayaraghavan, Duke Math. J., Volume 11 (1944), pp. 103-107 | Article | Zbl 0063.06657

[23] Raphael Salem Power series with integral coefficients, Duke Math. J., Volume 12 (1945), pp. 153-173 | Article | MR 11720 | Zbl 0060.21601

[24] Andrzej Schinzel Reducibility of polynomials and covering systems of congruences, Acta Arith., Volume 13 (1967), pp. 91-101 | Article | MR 219515 | Zbl 0171.00701

[25] Christopher J. Smyth Conjugate algebraic numbers on conics, Acta Arith., Volume 40 (1982), pp. 333-346 | Article | MR 667044 | Zbl 0431.12001

[26] Christopher J. Smyth Additive and multiplicative relations connecting conjugate algebraic numbers, J. Number Theory, Volume 23 (1986), pp. 243-254 | Article | MR 845906 | Zbl 0586.12001

[27] Christopher J. Smyth Salem numbers of negative trace, Math. Comput., Volume 69 (2000) no. 230, pp. 827-838 | Article | MR 1648407 | Zbl 0988.11050

[28] Annick Valibouze Sur les relations entre les racines d’un polynôme, Acta Arith., Volume 131 (2008) no. 1, pp. 1-27 | Article | MR 2373513 | Zbl 1137.12003

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