On the Degree of the GCD of Random Polynomials over a Finite Field

Abstract

In this paper, we focus on the degree of the greatest common divisor ( $\text{gcd}$ ) of random polynomials over ${\mathbb{F}}_q$ . Here, ${\mathbb{F}}_q$ is the finite field with $q$ elements. Firstly, we compute the probability distribution of the degree of the $\text{gcd}$ of random and monic polynomials with fixed degree over ${\mathbb{F}}_q$ . Then, we consider the waiting time of the sequence of the degree of $\text{gcd}$ functions. We compute its probability distribution, expectation, and variance. Finally, by considering the degree of a certain type $\text{gcd}$ , we investigate the probability distribution of the number of rational (i.e., in ${\mathbb{F}}_q$ ) roots (counted with multiplicity) of random and monic polynomials with fixed degree over ${\mathbb{F}}_q$ .