Abstract
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 .
1. Introduction
1.1. Background
The greatest common divisor function is very basic in number theory. It has also been considered in the view of probability theory. For an integer , suppose the random variables are independent and uniformly distributed on . In 1880s, Cesàro [1, 2] first considered the probability distribution of the of random integers and showed thatfor and , where is the Riemann zeta function. Diaconis and Erdös [3] gave a more precise asymptotic formula for the case , which isfor as . One may refer to [4–7], for more related works. In 2013, Fernández and Fernández [8] considered the waiting time for the sequence:
Let be the subscript at which the sequence reaches the value 1 for the first time. They computed the expectation of and showed that
Besides random integers, it is also natural to study random polynomials. One interesting topic in this area is to understand the behavior of the number of certain type of roots of random polynomials. For example, Kac [9] considered the number of real roots of random polynomials over the real number field . One may refer to [10] for a recent progress.
In this paper, we focus on the degree of the of random polynomials over the finite field , where is a prime power.
1.2. Our Results
In the polynomial ring , we use and to denote the sets of all monic polynomials and monic polynomials with degree , respectively. We also use to denote the degree of a polynomial .
For integers and , we definewhere , , are independent and uniformly distributed on . We derive the following probability distribution of .
Theorem 1. For any integers and , the mass function of iswhere
With the help of Theorem 1, we investigate the waiting time of the sequence:where are independent and uniformly distributed on , . Observe that this sequence is decreasing.
For an integer , define the random variable to be the subscript at which the sequence reaches a value not exceeding for the first time. We compute the probability distribution of and then derive its expectation and variance .
Theorem 2. Suppose integer ; then, for integers and , the mass function of isand furthermore, we have
It is a little bit surprising that the expectation and variance of are independent of the degree . Using SageMath, we verify this for and some by doing numerical experiments with times. The results are listed in Table 1 (expectation) and Table 2 (variance).
Enlightened by the proof of Theorem 1, we use the degree of to study the number of rational roots (counted with multiplicity) of a random polynomial , where is uniformly distributed on . Denote this number by ; then, we have the following result.
Theorem 3. For an integer , the mass function of isfor .
The method for proving Theorem 3 is also valid if we consider the number of distinct rational roots of a random polynomial . This number is investigated by Leont’ev in [11], where combinational methods are used. Comparatively, our method has more flavor of number theory, and we hope it can be used for other roots’ counting problems.
Notations: we use to denote the probability of an event and use and to denote the expectation and variance of a random variable . We also use to denote the finite field with elements and use to denote the polynomial ring over .
2. Preliminaries
The Mbius function for monic polynomials is defined by if is a product of distinct monic irreducible polynomials and if is not square free. For any , we have
For the mean value of over , it is well known that
For a polynomial in , we defined its norm by . We derive the following two results, which are needed in proving Proposition 1.
Lemma 1. For integers , we have
Proof. Note thatThen, the first statement follows by notingFor the second statement, we haveThis together with (13) gives our desired result.
The following lemma is used in the proof of Proposition 2.
Lemma 2. For integer , suppose and with and . Then, we have
Proof. Note thatLet ; then, by the definition of and , we have that is of the formfor some distinct , . From this, we derive thatNote that ; then, we haveThen, our required result follows by combining (21) and (22) with (19).
3. Proof of Theorem 1
We first compute the th power moments of .
Proposition 1. For any integers and , we havewhere is given by (7).
Proof. By the definition of the th moment, we haveIt follows thatThen, for the inner sum on the right-hand side of (25), we can writewhere we have used (12). Changing the order of the summations, we obtainIt follows thatInserting (28) to (25) givesThe contribution of those with is equal toBy Lemma 1, the contribution of those with is equal toHence, we havewhich is our desired result.
Now, we are ready to prove Theorem 1. Suppose is the moment generating function of ; then, we have
It follows from Proposition 1 that
Then, our desired result follows from the relationship between the moment generating function and the generating function of .
4. Proof of Theorem 2
For an integer , note that the event coincides with the event for each . Hence, by Theorem 1, we have
This gives the mass function in Theorem 2.
By the definition of the expectation of , we have
It follows from (35) that
To deal with , we write
For , we have
It follows from (35) again thatwhere we have used
Inserting (37) and (40) into (38) yields our desired result.
5. Proof of Theorem 3
We first compute the th power moments of .
Proposition 2. For any integers , we have
Proof. Proof. Let . Notice thatwhere is random and uniformly distributed on . Then, by the definition of the th moment, we haveIt follows thatFor the inner sum on the right-hand side of (45), we havewhere we have used (12). Changing the order of the summations, we deriveBy (45) and (47), we obtainBreaking the above sum into two sums according to or not, we havewhereTo deal with , note that and ; then, is of the formfor distinct , , where and . Thus, we haveFor , using Lemma 2, we deriveLet ; then, we havePlugging (52) and (54) into (49) yields our required result.
Now, we are ready to prove Theorem 3. Suppose is the moment generating function of ; then, we have
By Proposition 2, we derivewhich gives
Then, our desired result follows from the relationship between the moment generating function and the generating function of .
Data Availability
The date in the chart in our paper can be verifies by using SageMath.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
The first listed author was partially supported by National Natural Science Foundation of China (NSFC, Grant no. 12071238) and Shandong Provincial Natural Science Foundation, China (Grant no. ZR2019BA028). The second listed author was partially supported by National Natural Science Foundation of China (NSFC, Grant no. 12071238).