Glasnik Matematicki, Vol. 52, No. 2 (2017), 235-240.

ROOTS OF UNITY AS QUOTIENTS OF TWO CONJUGATE ALGEBRAIC NUMBERS

Artūras Dubickas

Department of Mathematics and Informatics, Vilnius University, Naugarduko 24, Vilnius LT-03225, Lithuania
e-mail: arturas.dubickas@mif.vu.lt

Abstract.   Let α be an algebraic number of degree d ≥ 2 over Q. Suppose for some pairwise coprime positive integers n1,… ,nr we have degnj) < d for j=1,…,r, where degn)=d for each positive proper divisor n of nj. We prove that then φ(n1 … nr) ≤ d, where φ stands for the Euler totient function. In particular, if nj=pj, j=1,…,r, are any r distinct primes satisfying degpj) < d, then the inequality (p1-1)… (pr-1) ≤ d holds, and therefore r ≪ log d/log log d for d ≥ 3. This bound on r improves that of Dobrowolski r ≤ log d/log 2 proved in 1979 and is best possible.

2010 Mathematics Subject Classification.   11R04, 11R18.

Key words and phrases.   Root of unity, conjugate algebraic numbers, degenerate linear recurrence sequence.


Full text (PDF) (access from subscribing institutions only)

DOI: 10.3336/gm.52.2.03


References:

  1. M. G. Aschbacher and R. M. Guralnick, On Abelian quotients of primitive groups, Proc. Amer. Math. Soc. 107 (1989), 89-95.
    MathSciNet     CrossRef

  2. J. Berstel and M. Mignotte, Deux propriétés décidables des suites recurrentes linéaires, Bull. Soc. Math. France 104 (1976), 175-194.
    MathSciNet     CrossRef

  3. E. Dobrowolski, On a question of Lehmer and the number of irreducible factors of a polynomial, Acta Arith. 34 (1979) 391-401.
    MathSciNet     CrossRef

  4. P. Drungilas and A. Dubickas, On subfields of a field generated by two conjugate algebraic numbers, Proc. Edinburgh Math. Soc. 47 (2004), 119-123.
    MathSciNet     CrossRef

  5. A. Dubickas, Roots of unity as quotients of two roots of a polynomial, J. Austral. Math. Soc. 92 (2012), 137-144.
    MathSciNet     CrossRef

  6. A. Dubickas and M. Sha, Counting degenerate polynomials of fixed degree and bounded height, Monatsh. Math. 177 (2015), 517-537.
    MathSciNet     CrossRef

  7. G. Everest, A. van der Poorten, I. Shparlinski and T. Ward, Recurrence sequences, Mathematical Surveys and Monographs 104, American Mathematical Society, Providence, RI, 2003.
    MathSciNet     CrossRef

  8. I. M. Isaacs, Quotients which are roots of unity (solution of problem 6523), Amer. Math. Monthly 95 (1988), 561-562.
    MathSciNet     CrossRef

  9. D. Masser, Auxiliary polynomials in number theory, Cambridge Tracts in Mathematics 207, Cambridge University Press, Cambridge, 2016.
    MathSciNet     CrossRef

  10. E. M. Matveev, On a connection between the Mahler measure and the discriminant of algebraic numbers, Math. Notes 59 (1996), 293-297.
    MathSciNet     CrossRef

  11. P. Robba, Zéros de suites récurrentes linéaires, Groupe Étude Anal. Ultramétrique, 5e Année (1977/78), Exposé No. 13, Paris, 1978, 5 p.
    MathSciNet    

  12. A. Schinzel, Around Pólya's theorem on the set of prime divisors of a linear recurrence, in: Diophantine equations. Tata Inst. Fund. Res. Stud. Math., 20, Tata Inst. Fund. Res., Mumbai, 2008, pp. 225-233.
    MathSciNet    

  13. K. Yokoyama, Z. Li and I. Nemes, Finding roots of unity among quotients of the roots of an integral polynomial, in: Proceedings of the 1995 international symposium on symbolic and algebraic computation (ed. A.H.M. Levelt), ISSAC'95, Montreal, Canada, July 10-12, 1995, New York, NY: ACM Press, 1995, pp. 85-89.

Glasnik Matematicki Home Page