Let $f(T)$ be a monic polynomial of degree $d$ with coefficients in a finite field $\mathbb{ F}_q$. Extending earlier results in the literature, but now allowing $(q,2d)>1$, we give a criterion for $f$ to satisfy the following property: for all but $d^2-d-1$ values of $s$ in $\mathbb{ F}_q$, the probability that $f(T)+sT+b$ is irreducible over $\mathbb{ F}_q$ (as $b\in\mathbb{ F}_q$ is chosen uniformly at random) is $1/d+O(q^{ -1/2})$.
Categories: math.AG