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Cubo (Temuco)

On-line version ISSN 0719-0646

Cubo vol.20 no.2 Temuco June 2018

http://dx.doi.org/10.4067/S0719-06462018000200067 

Articles

Some remarks on the non-real roots of polynomials

Shuichi Otake1 

Tony Shaska2 

1Waseda University, Department of Applied Mathematics, Japan. shuichi.otake.8655@gmail.com

2Oakland University, Department of Mathematics and Statistics, Rochester, MI, 48309.shaska@oakland.edu

ABSTRACT

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 and Phrases: Polynomials; non-real roots; discriminant; Bezoutian; Galois groups

RESUMEN

Sea f ∈ ℝ (t) dada por f(t, x) = xn + t · g(x) y β1 < · · · < βm las diferentes raíces reales del discriminante ∆(f,x)(t) de f(t, x) con respecto de x. Sea γ el número de raíces reales de . Para todo ξ > |βm|, si n − s es impar entonces el número de raíces reales de f(ξ, x) es γ + 1, y si n − s es par entonces el número de raíces reales de f(ξ, x) es γ, γ + 2 si ts > 0 o ts < 0, respectivamente. Un caso especial del resultado anterior es construyendo una familia de polinomios irreducibles sobre ℚ de grado n ≥ 3 con muchas raíces no-reales y grupo de automorfismos Sn

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