JournalsrlmVol. 28, No. 4pp. 747–775

On the regularity of the roots of a polynomial depending on one real parameter

  • Ferruccio Colombini

    Università di Pisa, Italy
  • Nicola Orrù

    Liceo Scientifico A. Pacinotti, Cagliari, Italy
  • Ludovico Pernazza

    Università di Pavia, Italy
On the regularity of the roots of a polynomial depending on one real parameter cover

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Abstract

We investigate the regularity of functions τ\tau of one variable such that P(t,τ(t))=0P(t,\tau(t))=0, where P(t,x)P(t,x) is a given polynomial of degree mm in xx whose coefficients are functions of class Cm2!C^{m^2!} of one real parameter. We show that if a root is chosen with a continuous dependence on the parameter, this function is indeed absolutely continuous. From this and a theorem of Kato one deduces that such polynomials have complete systems of roots that are absolutely continuous functions.

Cite this article

Ferruccio Colombini, Nicola Orrù, Ludovico Pernazza, On the regularity of the roots of a polynomial depending on one real parameter. Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. 28 (2017), no. 4, pp. 747–775

DOI 10.4171/RLM/784