Corrigendum to “Factorization type probabilities of polynomials with prescribed coefficients over a finite field” (Acta Arith. 194 (2020), 315–318)
Volume 224 / 2026
Acta Arithmetica 224 (2026), 95-97
MSC: Primary 14G15; Secondary 12F10
DOI: 10.4064/aa251204-28-5
Published online: 8 June 2026
Abstract
We correct a gap in the paper of the title. Let $f(T)$ be a monic polynomial of degree $d$ with coefficients in a finite field $\mathbb F_q$. 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 $f(T)+sT+b$ has a given factorization type $\lambda $ over $\mathbb F_q$, as $b\in \mathbb F_q$ is chosen uniformly at random, is $p_\lambda +O_d(q^{-1/2})$, where $p_\lambda $ is the probability that a permutation in $S_d$ has cycle shape $\lambda $. The assumptions allow $(q,2d) \gt 1$.
