Abstract
In this paper, we focus on the degree of the greatest common divisor ( gcd ) of random polynomials over F q . Here, 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 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 F q ) roots (counted with multiplicity) of random and monic polynomials with fixed degree over F q .
Highlights
We focus on the degree of the gcd of random polynomials over the finite field Fq, where q ≥ 2 is a prime power
In the polynomial ring Fq[T], we use M and Mn to denote the sets of all monic polynomials and monic polynomials with degree n ≥ 0, respectively
Enlightened by the proof of eorem 1, we use the degree of gcd to study the number of rational roots of a random polynomial f ∈ F q[T], where f is uniformly distributed on Mn, n ≥ 1
Summary
E method for proving eorem 3 is valid if we consider the number of distinct rational roots of a random polynomial f ∈ Mn. is number is investigated by Leont’ev in [11], where combinational methods are used. Enlightened by the proof of eorem 1, we use the degree of gcd to study the number of rational roots (counted with multiplicity) of a random polynomial f ∈ F q[T], where f is uniformly distributed on Mn, n ≥ 1. Denote this number by N(n, q); we have the following result. En, our required result follows by combining (21) and (22) with (19)
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