On the Degree of the GCD of Random Polynomials over a Finite Field - Kui Liu; Meijie Lu (Journal of Mathematics)
In this paper, we focus on the degree of the greatest common divisor (gcd) of random polynomials over Fq. Here, Fq is the finite field with q elements. Firstly, we compute the probability distribution of the degree of the gcd of random and monic polynomials with fixed degree over Fq. Then, we consider the waiting time of the sequence of the degree of gcd functions. We compute its probability distribution, expectation, and variance. Finally, by considering the degree of a certain type gcd, we investigate the probability distribution of the number of rational (i.e., in Fq) roots (counted with multiplicity) of random and monic polynomials with fixed degree over Fq.