# 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

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## Abstract

We investigate the regularity of functions $\tau$ of one variable such that $P(t,\tau(t))=0$, where $P(t,x)$ is a given polynomial of degree $m$ in $x$ whose coefficients are functions of class $C^{m^2!}$ 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