On the absolute continuity of the roots of some algebraic equations (Q2756134)

Let \(P(z,t)\equiv z^m+ a_1(t) z^{m- 1}+\cdots+ a_m(t)\) be a polynomial whose coefficients are functions defined on an open interval \(J\), and let functions \(z_k(t)\), be such that NEWLINE\[NEWLINEP(z, t)= \prod^m_{k= 1} (z- z_k(t))\quad\text{for all }z\text{ in }\mathbb{C}\text{ and }t\in J.NEWLINE\]NEWLINE It is known that if \(C_k(t)\in C^m(J)\) and all functions \(z_k(t)\) are real then one can select \(z_k(t)\) to be differentiable on \(J\).NEWLINENEWLINENEWLINEMain result of this paper (Theorem 5) is as follows. For every polynomial \(P(z,t)\) of degree \(\leq 3\) with coefficients in \(C^\infty(J)\) one can find absolutely continuous roots \(z_k(t)\).