Abstract

OTAKE, Shuichi  and  SHASKA, Tony. Some remarks on the non-real roots of polynomials. Cubo [online]. 2018, vol.20, n.2, pp.67-93. ISSN 0719-0646.  http://dx.doi.org/10.4067/S0719-06462018000200067.

Let f ∈ ℝ(t) be given by f(t, x) = xn + t · g(x) and β1 < · · · < βm the distinct real roots of the discriminant ∆(f,x)(t) of f(t, x) with respect to x. Let γ be the number of real roots of . For any ξ > |βm|, if n − s is odd then the number of real roots of f(ξ, x) is γ + 1, and if n − s is even then the number of real roots of f(ξ, x) is γ, γ + 2 if ts > 0 or ts < 0 respectively. A special case of the above result is constructing a family of degree n ≥ 3 irreducible polynomials over ℚ with many non-real roots and automorphism group Sn.

Keywords : Polynomials; non-real roots; discriminant; Bezoutian; Galois groups.

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