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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
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
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
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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