Rad HAZU, Matematičke znanosti, Vol. 21 (2017), 21-27.

ROOT SEPARATION FOR REDUCIBLE MONIC POLYNOMIALS OF ODD DEGREE

Andrej Dujella and Tomislav Pejković

Department of Mathematics, Faculty of Science, University of Zagreb, 10000 Zagreb, Croatia
e-mail: duje@math.hr

Department of Mathematics, Faculty of Science, University of Zagreb, 10000 Zagreb, Croatia
e-mail: pejkovic@math.hr


Abstract.   We study root separation of reducible monic integer polynomials of odd degree. Let H(P) be the naïve height, sep(P) the minimal distance between two distinct roots of an integer polynomial P(x) and sep(P) = H(P)-e(P). Let er*(d) = lim supdeg(P)=d, H(P)→+∞ e(P), where the lim sup is taken over the reducible monic integer polynomials P(x) of degree d. We prove that er*(d) ≤ d - 2. We also obtain a lower bound for er*(d) for d odd, which improves previously known lower bounds for er*(d) when d ∈ {5, 7, 9}.

2010 Mathematics Subject Classification.   11C08, 12D10, 11B37.

Key words and phrases.   Integer polynomials, root separation.


Full text (PDF) (free access)

DOI: http://doi.org/10.21857/mnlqgcj04y


References:

  1. Y. Bugeaud, Approximation by algebraic numbers, Cambridge Tracts in Mathematics 160, Cambridge University Press, Cambridge, 2004.
    MathSciNet     CrossRef

  2. Y. Bugeaud and A. Dujella, Root separation for irreducible integer polynomials, Bull. Lond. Math. Soc. 43 (2011), 1239-1244.
    MathSciNet     CrossRef

  3. Y. Bugeaud and A. Dujella, Root separation for reducible integer polynomials, Acta Arith. 162 (2014), 393-403.
    MathSciNet     CrossRef

  4. Y. Bugeaud, A. Dujella, T. Pejković and B. Salvy, Absolute real root separation, Amer. Math. Monthly, to appear.

  5. Y. Bugeaud and M. Mignotte, Polynomial root separation, Intern. J. Number Theory 6 (2010), 587-602.
    MathSciNet     CrossRef

  6. A. Dujella and T. Pejković, Root separation for reducible monic quartics, Rend. Semin. Mat. Univ. Padova 126 (2011), 63-72.
    MathSciNet     CrossRef

  7. J.-H. Evertse, Distances between the conjugates of an algebraic number, Publ. Math. Debrecen 65 (2004), 323-340.
    MathSciNet

  8. K. Mahler, An inequality for the discriminant of a polynomial, Michigan Math. J. 11 (1964), 257-262.
    MathSciNet

  9. T. Pejković, P-adic root separation for quadratic and cubic polynomials, Rad Hrvat. Akad. Znan. Umjet. Mat. Znan. 20 (2016), 9-18.
    MathSciNet

  10. A. Schönhage, Polynomial root separation examples, J. Symbolic Comput. 41 (2006), 1080-1090.
    MathSciNet     CrossRef

  11. N. P. Smart, The Algorithmic Resolution of Diophantine Equations, Cambridge University Press, Cambridge, 1998.
    MathSciNet     CrossRef


Rad HAZU Home Page