Rad HAZU, Matematičke znanosti, Vol. 20 (2016), 9-18.

P-ADIC ROOT SEPARATION FOR QUADRATIC AND CUBIC POLYNOMIALS

Tomislav Pejković

Abstract.   We study p-adic root separation for quadratic and cubic polynomials with integer coefficients. The quadratic and reducible cubic polynomials are completely understood, while in the irreducible cubic case and p ≠ 2, we give a family of polynomials with the bound which is the best currently known.

2010 Mathematics Subject Classification.   11C08, 11B37, 11J61.

Key words and phrases.   Integer polynomials, root separation, p-adic numbers.

References:

1. E. Bombieri and W. Gubler, Heights in Diophantine geometry, New Mathematical Monographs, 4. Cambridge University Press, Cambridge, 2006.
   MathSciNet     CrossRef

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

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

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

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. F. Q. Gouvêa, p-adic numbers. An introduction, Second edition, Universitext, Springer- Verlag, Berlin, 1997.
   MathSciNet     CrossRef

9. N. Koblitz, p-adic numbers, p-adic analysis, and zeta-functions, Second edition, Graduate Texts in Mathematics, 58. Springer-Verlag, New York, 1984.
   MathSciNet     CrossRef

10. S. Lang, Algebra, Revised third edition, Graduate Texts in Mathematics, 211. Springer-Verlag, New York, 2002.
    MathSciNet     CrossRef

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

12. M. Mignotte, Mathematics for computer algebra, Springer-Verlag, New York, 1992.
    MathSciNet     CrossRef

13. J. F. Morrison, Approximation of p-adic numbers by algebraic numbers of bounded degree, J. Number Theory 10 (1978), 334-350.
    MathSciNet     CrossRef

14. T. Pejković, Polynomial Root Separation and Applications, PhD Thesis, Université de Strasbourg, Strasbourg, 2012.

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