Linear relations with conjugates of a Salem number

Dubickas, Artūras; Jankauskas, Jonas

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 $\alpha$. Nous montrons qu’une telle relation provient d’une relation linéaire entre les conjugués de l’entier algébrique totalement réel correspondant $\alpha +1/\alpha$. 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 $\alpha$. We show that every such a relation arises from a linear relation between conjugates of the corresponding totally real algebraic integer $\alpha +1/\alpha$. 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 $6$.

Classification : 11R06, 11R09
Keywords: linear additive relations, Salem numbers, Pisot numbers, totally real algebraic numbers
Mots clés : Les relations linéaires additives, les nombres de Salem, les nombres de Pisot, les nombres algébriques totalement réels

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

[2] Beukers, Frits; Smyth, Christopher J. 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

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

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

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

[6] Drmota, Michael; Skałba, Mariusz 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

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

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

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

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

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

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

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

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

[15] Girstmair, Kurt 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 | DOI | MR | Zbl

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

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

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

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

[20] Lalande, Franck 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 | Zbl

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

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

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

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

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

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

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

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