On the degree of the GCD of random polynomials over a finite field
Summary: In this paper, we focus on the degree of the greatest common divisor (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 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 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 \(\mathbb{F}_q\)) roots (counted with multiplicity) of random and monic polynomials with fixed degree over \(\mathbb{F}_q\).
scientific article; zbMATH DE number 7433534