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: 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 e_r*(d) = lim sup_{deg(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 e_r*(d) ≤ d - 2. We also obtain a lower bound for e_r*(d) for d odd, which improves previously known lower bounds for e_r*(d) when d ∈ {5, 7, 9}.

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

Key words and phrases.   Integer polynomials, root separation.

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

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