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

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