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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Autores principales:	Otake,Shuichi, Shaska,Tony
Lenguaje:	English
Publicado:	Universidad de La Frontera. Departamento de Matemática y Estadística. 2018
Materias:	Polynomials non-real roots discriminant Bezoutian Galois groups
Acceso en línea:	http://www.scielo.cl/scielo.php?script=sci_arttext&pid=S0719-06462018000200067
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spelling	oai:scielo:S0719-064620180002000672019-02-13Some remarks on the non-real roots of polynomialsOtake,ShuichiShaska,Tony Polynomials non-real roots discriminant Bezoutian Galois groups 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.info:eu-repo/semantics/openAccessUniversidad de La Frontera. Departamento de Matemática y Estadística.Cubo (Temuco) v.20 n.2 20182018-06-01text/htmlhttp://www.scielo.cl/scielo.php?script=sci_arttext&pid=S0719-06462018000200067en10.4067/S0719-06462018000200067
institution	Scielo Chile
collection	Scielo Chile
language	English
topic	Polynomials non-real roots discriminant Bezoutian Galois groups
spellingShingle	Polynomials non-real roots discriminant Bezoutian Galois groups Otake,Shuichi Shaska,Tony Some remarks on the non-real roots of polynomials
description	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.
author	Otake,Shuichi Shaska,Tony
author_facet	Otake,Shuichi Shaska,Tony
author_sort	Otake,Shuichi
title	Some remarks on the non-real roots of polynomials
title_short	Some remarks on the non-real roots of polynomials
title_full	Some remarks on the non-real roots of polynomials
title_fullStr	Some remarks on the non-real roots of polynomials
title_full_unstemmed	Some remarks on the non-real roots of polynomials
title_sort	some remarks on the non-real roots of polynomials
publisher	Universidad de La Frontera. Departamento de Matemática y Estadística.
publishDate	2018
url	http://www.scielo.cl/scielo.php?script=sci_arttext&pid=S0719-06462018000200067
work_keys_str_mv	AT otakeshuichi someremarksonthenonrealrootsofpolynomials AT shaskatony someremarksonthenonrealrootsofpolynomials
_version_	1714206800270589952

Some remarks on the non-real roots of polynomials

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