On an estimation of polynomial roots by Lagrange

Maurice Mignotte (1) , Doru Stefanescu

1 IRMA - Institut de Recherche Mathématique Avancée

"Lagrange stated the following inequality: An upper bound forthe positive real roots of a monic polynomial over $R$ is equal to$R+ho$, where $R$ and $ho$ are the two largest numbers in the sethbox{${sqrt[j]{|a_j|}; ,jin J}$, where ${a_j; ,jin J}$ }are the negative coefficients. Since Lagrange does not provide a complete proof, wegive one following Cauchy's method. We also present a slightgeneralization of this theorem of Lagrange, using a result ofKojima."

Mots clés: Polynomial roots; bound of Lagrange; "Polynomial roots; bound of Lagrange"

Domaines: Variables complexes [math.CV]

Maurice Mignotte, Doru Stefanescu. On an estimation of polynomial roots by Lagrange. 2002. ⟨hal-00129675⟩