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Take a random polynomial $f$ with integer coefficients (e.g., choose coefficients between $1$ to $B$ of a fixed degree $n$ and let $B$ tend to $\infty$). Using computer we noted that the 2-valuation ...

Let $q$ be a power of a prime $p$. For every $i\in\mathbb N$, one denotes $D_i=\prod_{\substack{h\in\mathbb F_q[T]\text{ monic}\\\deg h=i}}\limits h$. For $n\in\mathbb N$, write $$n=n_0+n_1q+\cdots+...