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

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.

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.

Received:
Revised:
Accepted:
Published online:
DOI: 10.5802/jtnb.1116
Classification: 11R06,  11R09
Keywords: linear additive relations, Salem numbers, Pisot numbers, totally real algebraic numbers
Artūras Dubickas 1; Jonas Jankauskas 2

1 Institute of Mathematics Faculty of Mathematics and Informatics Vilnius University Naugarduko 24 03225 Vilnius, Lithuania
2 Mathematik und Statistik Montanuniversität Leoben Franz Josef Strasse 18 8700 Leoben, Austria
@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/}
}
TY  - JOUR
TI  - Linear relations with conjugates of a Salem number
JO  - Journal de Théorie des Nombres de Bordeaux
PY  - 2020
DA  - 2020///
SP  - 179
EP  - 191
VL  - 32
IS  - 1
PB  - Société Arithmétique de Bordeaux
UR  - https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1116/
UR  - https://doi.org/10.5802/jtnb.1116
DO  - 10.5802/jtnb.1116
LA  - en
ID  - JTNB_2020__32_1_179_0
ER  - 
%0 Journal Article
%T Linear relations with conjugates of a Salem number
%J Journal de Théorie des Nombres de Bordeaux
%D 2020
%P 179-191
%V 32
%N 1
%I Société Arithmétique de Bordeaux
%U https://doi.org/10.5802/jtnb.1116
%R 10.5802/jtnb.1116
%G en
%F JTNB_2020__32_1_179_0
Artūras Dubickas; Jonas Jankauskas. Linear relations with conjugates of a Salem number. Journal de Théorie des Nombres de Bordeaux, Volume 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

Cited by Sources: