Factorization type probabilities of polynomials with prescribed coefficients over a finite field

Abstract

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 $1/d+O(q^{-1/2})$.

A major error in the statement and proof of the main theorem from the previous version has been corrected, along with a number of further minor improvements

References in corpus (1)

• Prime and Möbius correlations for very short intervals in ${\mathbb{F}}_q[x]$