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

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.