On the regularity of the roots of a polynomial depending on one real parameter (Q1687603)

Summary: 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.