On the density of irreducible polynomials which generate k-free polynomials over function fields

https://doi.org/10.1016/j.ffa.2021.101979Get rights and content

Abstract

Let MFq[t] be a polynomial, and let k2 be an integer. In this note, we will compute the asymptotic density of irreducible monic polynomials PFq[t] for which P+M is not divisible by the kth power of any irreducible polynomial.

Introduction

In 1949, Mirsky [6] proved that every sufficiently large number can be written as a sum of a prime and a k-free number for any given k2. In fact, he derived an asymptotic formula for the number of such representations. In 2001, Yao [9] attempted to show that an analogous result holds in the case of function fields over finite fields. Apparently, according to Math Reviews (MR1841909), there seem to be errors in the paper. This was highlighted as an open problem in Moree's survey [7]. The purpose of this paper is to correct the error in [9] and establish the analogue in the function field case.

Let Fq be a finite field of order q, and let MFq[t] be a fixed polynomial. For k2, a polynomial is k-free if it is not divisible by the kth power of any irreducible polynomial. In this note, we want to compute the asymptotic density of irreducible polynomials P for which P+M is k-free. For the most part, we adhere to the notation and arrangement of [9]. More precisely, we will prove the following:

Theorem 1

Let MFq[t] be a fixed polynomial, and let k2 be an integer. Denote by P+ the set of all monic irreducible polynomials. We define the setUk(M,d)={PP+|degP=d,P+Mis k-free}. Then we havelimd#Uk(M,d)#{PP+|degP=d}=PMPP+(11|P|k|P|k1),where |P|=qdegP.

Theorem 1 can be directly obtained from the following result and the prime number theorem for Fq[t]:

Theorem 2

Let MFq[t] be a fixed polynomial, and let k2 be an integer. If d>degM, then#Uk(M,d)=qddPMPP+(11|P|k|P|k1)+O(qd2+d+2(1k)2k).

It may be possible to improve the error term and we hope to address this question in future work.

Section snippets

Preliminaries

We set the following notations throughout this note:A=Fq[t]A+P+The polynomial ring with one variable.The set of all monic polynomials in A.The set of all monic irreducible polynomials in A. For any QA+, we write its factorization Q=i=1rPini with PiP+. We define the polynomial Möbius function asμ(Q)={1if Q=1,0ifQis not square-free,(1)rifQ is square-free and Q=P1Pr, where PiP+. Analogously, for any integer k2, we defineμk(Q)={1ifQisk-free,0otherwise. It is easy to verify identityμk(Q)=aak

Proof of Theorem 2

We write#Uk(M,d)=degP=dPP+μk(P+M). Since we assumed d>degM, using (2.2), we have#Uk(M,d)=degP=dPP+akb=P+Mμ(a)=degad/k(a,M)=1μ(a)degP=dakb=P+MPP+1. For any td/k, we can split the sum as#Uk(M,d)=degat(a,M)=1μ(a)degP=dPM(mod ak)PP+1+t<degad/k(a,M)=1μ(a)degP=dPM(mod ak)PP+1. We will choose later a suitable t with 0<t<d/k to minimize the error term (the second term) of (3.2). This is how we fix the mistake in [9], which occurs on lines 10-13, where μ(a) should

Future directions

As noted earlier, it would be interesting to see if the error term in Theorem 2 can be improved. A generalization of Theorem 1 involving higher dimensional tuples of fixed polynomials can be discussed analogously following the generalization by Mirsky [4] for number fields. Moreover, further generalization is possible following the subsequent result of Mirsky [5].

Acknowledgement

We would like to thank Michael Rosen, Joseph H. Silverman, Wei-Chen Yao, and the anonymous referee for helpful suggestions on an earlier version of this paper.

References (9)

  • Wei-Chen Yao

    On an elementary density problem for polynomials over finite fields

    Finite Fields Appl.

    (2001)
  • Leonard Carlitz

    The arithmetic of polynomials in a Galois field

    Am. J. Math.

    (1932)
  • Gove W. Effinger et al.

    Additive Number Theory of Polynomials over a Finite Field

    (1991)
  • Heinrich Kornblum et al.

    Über die Primfunktionen in einer arithmetischen Progression

    Math. Z.

    (1919)
There are more references available in the full text version of this article.

Cited by (0)

1

Research of the first author partially supported by a Coleman Postdoctoral Fellowship.

2

Research of the second author partially supported by NSERC Discovery grant.

View full text