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Some remarks on the non-real roots of polynomials

Shuichi Otake, Tanush Shaska

I-AIMS paper 2017-3 · Published in Cubo 20 (2018), 67–93 · 2018

Let f ∈ ℝ(t)[x] be given by f(t,x) = xⁿ + t · g(x) and λ₁ < ⋯ < λₘ the distinct real roots of the discriminant Δ_((f,x))(t) of f(t,x) with respect to x. Let γ be the number of real roots of g(x) = ∑ₖ₌₀ˢ tₛ₋ₖ xˢ⁻ᵏ. For any ξ > |λₘ|, 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 or 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.

MSC classes: Primary 12D10; Secondary 12F10, 26C10

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