Let $P\in\mathbb F_q[X]$ be an irreducible polynomial, $Q\in\mathbb F_q[X]$. Do we have $v_P(Q(X^q))\ge v_P(Q)$ where $v_P$ denotes the $P$-valuation?
Obviously, if a root $x_0$ of $P$ is a root of $Q$, then $x_0$ is a root of $Q(X^q)$ too, but I do not manage to give a lower bound for its multiplicity.
