Some remarks on the non-real roots of polynomials
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 > |β...
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| Auteurs principaux: | , |
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| Langue: | English |
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Universidad de La Frontera. Departamento de Matemática y Estadística.
2018
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| Accès en ligne: | http://www.scielo.cl/scielo.php?script=sci_arttext&pid=S0719-06462018000200067 |
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| Résumé: | 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. |
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