Factorization type probabilities of polynomials with prescribed coefficients over a finite field
Volume 194 / 2020
Acta Arithmetica 194 (2020), 315-318
MSC: Primary 14G15; Secondary 12F10.
DOI: 10.4064/aa190420-31-10
Published online: 18 March 2020
Abstract
Let $f(T)$ be a monic polynomial of degree $d$ with coefficients in a finite field $\mathbb F _q$, having a nonconstant derivative and a nonzero second Hasse derivative. We prove that for all but $d^2-d-1$ values of $s\in \mathbb F _q$, the following holds: as $b\in \mathbb F _q$ is chosen uniformly at random, the probability that $f(T)+sT+b$ is irreducible is $1/d+O_d(q^{-1/2})$ as $q\to \infty $. In particular, as $s$ and $b$ are chosen uniformly at random in $\mathbb F _q$, the probability that $f(T)+sT+b$ is irreducible is $1/d+O_d(q^{-1/2})$ as $q\to \infty $.
