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More precisely, there exists a positive integer $k$ and a rational number $p >1$, both depending only on the degree $n$, such that if $a_j \\in C^{k}$ then any continuous choice of roots of $P_a$ is absolutely continuous with derivatives in $L^q$ for all $1 \\le q < p$, in a uniform way with respect to $\\max_j\\|a_j\\|_{C^k}$. 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