# Evaluation of Some Sums of Polynomials inFq[t]

Authors: Adama Diene
Journal: Journal of Applied Mathematics
 (2019)

---

## Abstract

We prove the polynomial analogues of some Liouville identities from elementary number theory. Consequently several sums defined over the finite fieldsFq[t] are evaluated by combining the results obtained and some of the results from sums of reciprocals of polynomials over Fq[t].


---

## Body

## 1. Introduction and Background

Letp be a prime number, q=pe for some positive integer e and Fq the finite field with q elements. It is well known that Fq[t] and the set of all integers Z share many similarities. Analogue of many results on Z has been proven for Fq[t]. Both rings are principal ideal domains, both have the property that the residue class ring of any nonzero ideal is finite, both rings have infinitely many prime elements, and both rings have finitely many units. For the ring Z, the units are ±1 and every nonzero integer is a multiple by a unit of a positive integer. Similarly, the units of Fq[t] are the nonzero scalars Fq∗, and every nonzero polynomial in Fq[t] is a multiple by a unit of a monic polynomial. Thus, one may think that many other results, which hold for Z, have their analogues in the ring Fq[t]. This is indeed the case. For example, analogues of the little theorem of Fermat and Euler, Wilsons theorem, quadratic (and higher) reciprocity, the prime number theorem, and Dirichlet’s theorem on primes in arithmetic progression and many more well-known theorems from elementary number theory have been proven true for Fq[t], see [1].In the nineteenth century, Joseph Liouville introduced a powerful new method into elementary number theory that allowed him to get many interesting identities. His approach was also used to solve well-known problems such as sums of squares, sums of triangular numbers, recurrence relations for divisor functions, convolution sums, and many others. Most of these problems are still today a subject of mathematical research.In this present paper, we propose to study the polynomial analogue of some identities of Liouville in Williams [2, chapter 3]. More precisely, if d(n) and σ(n) represent respectively the number of divisors and the sum of divisors of n, and(1)dk,mg≔∑d∈Nd∣gd≡kmod⁡m1,n∈N,Liouville proved the following.Theorem 1 (Liouville).
Leta∈Z∗, b∈Z, m∈N, and n∈N. Set(2)g≔gcda,m,a1≔ag,m1≔mg.If a2∈1,2,…,m1-1 satisfies (3)a1a2≡1mod⁡m1,then(4)∑d∈Nd∣nad+bm=amσn+bmdn-1m∑l=1l≡bmod⁡mm-1lda2l-b/g,m1n,where [x] represents the integer n satisfying n≤x<n+1.He also proved many other results related to Theorem1.The main goal of this paper is to evaluate some sums of polynomials overFq involving the polynomial analogue of Theorem 1. We also evaluate the average of some functions defined on the set of all monic polynomials over Fq.Throughout this paper, we letA=Fqt. We define M, Mn, and M<n to be the subset of all monic polynomials in A, the subset of all monic polynomials in A of degree n, and the subset of all monic polynomials in A of degree less than n, respectively. We also let Pn and P<n to be respectively the subset of all monic irreducible polynomials in A of degree n and the subset of all monic irreducible polynomials in A of degree less than n.Forf∈A, we denote the degree of f by deg⁡f, the number of polynomials g coprime to f with deg⁡g<deg⁡f by ϕ(f), the number of divisors of f by d(f) and we set(5)σkf=∑d∈Md∣fdkFinally, we setf=qdeg⁡f if f≠0 and f=0 if f=0. It is well known (see [1]) that every f∈A, f≠0 can be written uniquely in the form (6)f=αp1e1⋯pses,whereα is a nonzero element in Fq, p1,…,ps are irreducible monic polynomials, pi≠pj for i≠j, and e1,…,es are nonnegative integers.The following 2 propositions will be needed later. Their proofs are omitted here and can be found in [1].Proposition 2.
Letf∈A. If the prime decomposition of f is given by
f = α p 1 e 1 … p s e s, then(7)ϕf=f∏i=1s1-1pi.df=e1+1e2+1⋯es+1.σf=p1e1+1-1p1-1p2e2+1-1p2-1⋯pses+1-1ps-1.Proposition 3.
(8) ∑ f ∈ M n d f = n + 1 q n . ∑ f ∈ M n σ f = q 2 n · 1 - q - n - 1 1 - q - 1 .


## 2. Some Results about Sum of Reciprocals of Polynomials overFq[t]

In this section, we give a closed formula for some sums of reciprocals of polynomials overFq[t]. These results will be used in the subsequent sections to help prove some polynomial identities in Fq[t].We start with the following definitions. Lettingn∈N, we define dn and ln as(9)dn=∏i=0n-1tqn-tqi,ln=∏i=1nt-tqiFurther, we setd0=l0=1, and, for an integer k, we define(10)Snk=∑a∈Mn1ak,S<nk=∑a∈M<n1akThe next 2 theorems are due to Carlitz [3]. The first one provides an explicit formula for Sn(k) and S<n(k) if k=±(qi-1). Details of this theorem can be also found in [4].Theorem 4 (Carlitz [3]).
For eachn∈N, we have(11)Snqi-1=ln+i-1li-1lnqi,S<nqi-1=ln+i-1liln-1qi,(12)fori≥n,Sn-qi-1=dilndi-nqn,S<n-qi-1=di-1qln-1di-nqnThe next theorem is initially due to Carlitz [3], but an alternative proof was given by Hicks et al. [5] in a most recent paper published in 2012.Theorem 5 (Carlitz [3]).
Fork∈N, 1≤k≤q, we have (13)∑f∈Mn1fk=1∏i=1nt-tqik=1lnk.More results about sum of reciprocals of polynomials can be also found in [6]. We state here some of them. Let Rn be the subset of all monic reducible polynomials in A of degree less than n and we denote by hn the product of all monic irreducible polynomials of degree n in A, i.e., (14)hn=∏p∈Pnp.S. Nelson [6] in 2016 has proved the following theorem.Theorem 6 (Nelson [6]).
WithPn, Rn, and hn defined as above, we have (15)∑p∈P21p=-tq-tq-3h2t.∑p∈P31p=h2t3tq2-tq-4h3t.∑r∈R21r=1tq-t2.∑r∈R31r=-1tq2-ttq-t2.We conclude this section with our main theorem that states the following.Theorem 7 (main theorem).
Letm∈Mn, 0≠a∈Fq[t], and b∈M<n. Assume a=ms1+r1 and bu=ms2+r2, where ri∈M<n,i=1,2. Define(16)vf=∑b∈M<n∑d∈Md∣fdbu+amk,v1f=∑b∈M<n∑d∈Md∣fdbu+amk.If u=(m-1)mk-1, then(17)∑f∈Mnvf=dn-1mk-1mkln-1mk-1d0mkq2n·1-q-n-11-q-1-n+1qn1-m1-qmka-r1-1mk∑b∈M<n∑r2∈M<nbm-1=ms2+r2r2,and(18)∑f∈Mnv1f=amkqn1-qn1-qn+11-q2-n+1qndn-1qmkln-1.Theorem7 will be used in Section 4 to establish some polynomials identities in Fq. These identities are obtained by evaluating certain sums taken over the subset of all monic polynomials and monic irreducible polynomials.


## 3. Polynomial Analogue of Some Liouville’s Identities

Letk∈A and m∈M. We define the function dk,m(g) as the number of divisors d of g congruent to k modulo m, i.e.,(19)dk,mg≔∑d∈Md∣gd≡kmod⁡m1,g∈M,The following theorem is the polynomial analogue of [2, theorem 3.8]. It describes some of the properties of dk,m for polynomials. The proof follows the same steps as the one for integers proved by Liouville in [2].Lemma 8.
Form,g∈M, we have(20)d0,mg=dgm,∑k∈Mdeg⁡k<deg⁡mdk,mg=dg.Proof.
The proof is essentially the same as the integer case. We have(21)d0,mg=∑d∈Md∣gd≡0mod⁡m1=∑e∈Mem∣g1=∑e∈Me∣g/m1=dgm.For m=1, we have (22)d0,1g=dg.
For the second equality, we have(23)∑k∈Mdeg⁡k<deg⁡mdk,mg=∑k∈Mdeg⁡k<deg⁡m∑d∈Md∣gd≡kmod⁡m1=∑d∈Md∣g1=dg.Letm,a∈Fq[t]. The division algorithm theorem states that that there exists a unique set of polynomials s,r∈Fq[t] satisfying a=ms+r, with r=0 or deg⁡r<deg⁡m. We define (24)am=s.Remark 9.
Since fora,b∈Fq[t], deg⁡(a+b) is the maximum between deg⁡a and deg⁡b, for m∈M, we have (25)a+bm=am+bm.Lemma 10 (see [2]).
Let0≠a,b∈Fq[t] and f,m∈M. Then(26)∑b∈Mdeg⁡b≤deg⁡m∑d∈Md∣fdam=1-m1-qmaσf.Proof.
We have(27)∑k∈Mdeg⁡b≤deg⁡m∑d∈Md∣fdam=am∑b∈Mdeg⁡b≤deg⁡m∑d∈Md∣fd=am∑b∈Mdeg⁡b≤deg⁡mσf=amσf∑b∈Mdeg⁡b≤deg⁡m1=amσf∑i=0deg⁡mqi=1-m1-qmaσf.The next theorem is the polynomial analogue of [2, theorem 3.9].Theorem 11.
Let0≠a,b∈Fq[t] and m∈M. If gcd(a,m)=1 and aa1≡1modm, then(28)∑d∈Md∣fda+bm=amσf+bmdf-1m∑r∈Mdeg⁡r<deg⁡mrda1r-b,m.Proof.
For eachd∈M, there exist a unique set of polynomials s,r∈M such that da+b=ms+r,deg⁡r<deg⁡m. Therefore, we have(29)∑d∈Md∣fda+bm=∑s∈M∑d∈Md∣fda+b/m=ss=∑r∈Mdeg⁡r<deg⁡m∑d∈Md∣fda+b=ms+rda+b-rm=∑d∈Md∣fda+bm-∑r∈Mdeg⁡r<deg⁡m∑d∈Md∣fda+b=ms+rrm=am∑d∈Md∣fdd+km∑d∈Md∣f1-1m∑r∈Mdeg⁡r<deg⁡mr∑d∈Md∣fda≡r-bmodm1=am∑d∈Md∣fd+bm∑d∈Md∣f1-1m∑r∈Mdeg⁡r<deg⁡mr∑d∈Md∣fd≡h1r-bmodm1=amσf+bmdf-1m∑r∈Mdeg⁡r<deg⁡mrda1r-b,mf.Our last polynomial analogue result in this section deals with the functionFd(a) defined for f,d∈M as(30)Fdf=1,ifd∣f0,ifd∤f.Lemma 12.
For anyd1∈M1,(31)∑f∈Mn+1Fd1f=∑b∈M<n1.Proof.
We have(32)∑f∈Mn+1Fd1f=∑f∈Mn+1d1∣f1+∑f∈Mn+1d1∤f0=∑f∈M1d1∣f1+∑f∈M2d1∣f1+⋯+∑f∈Mn+1d1∣f=1+∑f∈M2e∈M1f=d1×e1+∑f∈M3e∈M2f=d1×e1+⋯+∑f∈Mn+1e∈Mnf=d1×e1=1+q+q2+⋯+qn=∑b∈M<n1.


## 4. Proof of the Main Theorem

In this section, we prove the main Theorem7 stated in Section 2. Some other sums are also evaluated. We need the following lemma.Lemma 13 (see [4]).
Let0≠a,b∈Fq[t] and f,m∈M. Then(33)∑b∈Mdeg⁡b<deg⁡m∑d∈Md∣fdbm-1mk-1mk=dn-1mk-1ln-1mk-1d0mkσfmk.Proof.
We have(34)∑b∈Mdeg⁡b<deg⁡m∑d∈Md∣fdbm-1mk-1mk=1mk∑b∈Mdeg⁡b<deg⁡mbm-1mk-1∑d∈Md∣fd=σfmk∑b∈Mdeg⁡b<deg⁡mbm-1mk-1=σfmk∑b∈Mdeg⁡b<deg⁡mbm-1mk-1.
Therefore, by Theorem4 the proof is complete.


### 4.1. Proof of the Main Theorem

Proof.
We havedbu+a=(ds2+s1)mk+(dr2+r1) and(35)dbu+amk=dbumk+amk.
Therefore, we have(36)vf=∑b∈M<n∑d∈Md∣fdbu+amk=∑b∈M<n∑d∈Md∣fdbumk+∑b∈M<n∑d∈Md∣famk=∑b∈M<n∑s2∈M∑d∈Md∣fdbu/mk=ds2ds2+∑b∈M<n∑s1∈M∑d∈Md∣fa/mk=s1s1
Rewriting the summation in terms ofr1 and r2 yields(37)vf=∑b∈M<n∑d∈Md∣fa=ms1+r1a-r1mk+∑b∈M<n∑r2∈M<n∑d∈Md∣fbu=ms2+r2dbu-r2mk.
Hence Lemmas10 and 13 imply(38)vf=1-m1-qmka-r1df+dn-1mk-1ln-1mk-1d0mkσfmk-dfmk∑b∈M<n∑r2∈M<nbm-1=ms2+r2r2=dn-1mk-1ln-1mk-1d0mkσfmk+df1-m1-qmka-r1-1mk∑b∈M<n∑r2∈M<nbm-1=ms2+r2r2.
Therefore,(39)∑f∈Mnvf=dn-1mk-1mkln-1mk-1d0mk∑f∈Mnσf-1-m1-qmka-r1-1mk∑b∈M<n∑r2∈M<nbu=mks2+r2r2∑f∈Mndf=dn-1mk-1mkln-1mk-1d0mkq2n·1-q-n-11-q-1-n+1qn1-m1-qmka-r1-1mk∑b∈M<n∑r2∈M<nbu=mks2+r2r2.
The proof of (18) is similar and is a direct consequence of Lemma 13.Corollary 14.
Ifm=p∈Pn, then(40)∑f∈Mnvf=dn-1pk-1ln-1pk-1d0pkpkq2n·1-q-n-11-q-1-n+1qn1-p1-qpka-r1-1pk1-p1-q.Proof.
If∈Pn, then, for each b∈M<n,bu=pks2+r2, we have (41)bu=bp-1pk-1≡r2mod⁡pk,which forces r2=1 by Euler theorem for polynomial; see [1]. Therefore,(42)∑b∈M<n∑r2∈M<nbp-1=ps2+r2r2=1-p1-q,and the corollary follows from the main theorem.


### 4.2. Evaluation of Some Sums inFq[t]

Theorem 15.
Letn∈N and for k∈N.
If1≤k≤q, then(43)∑m∈Mn∑f∈Mnv1f=aqn1-qn1-qn+11-q2-n+1qndn-1qln-11lnk.Proof.
Using the main theorem, we obtain(44)∑m∈Mn∑f∈Mnv1f=∑m∈Mnamkqn1-qn1-qn+11-q2-n+1qndn-1qmkln-1.
Therefore, if1≤k≤q, Theorem 5 implies(45)∑m∈Mnamkqn1-qn1-qn+11-q2-n+1qndn-1qmkln-1=aqn1-qn1-qn+11-q2-n+1qndn-1qln-11lnk.
which complete the proof of (43).Theorem 16.
Fork=1, we have (46)∑m∈P2∑f∈Mnv1f=atq-tq-3h2tqn1-qn1-qn+11-q2+n+1qndn-1qln-1tq-tq-3h2t.(47)∑m∈P3∑f∈Mnv1f=ah2t3tq2-tq-4h3tqn1-qn1-qn+11-q2-n+1qndn-1qln-1h2t3tq2-tq-4h3t.Proof.
By the main theorem, we have(48)∑m∈P2∑f∈Mnv1f=∑m∈P2amkqn1-qn1-qn+11-q2-n+1qndn-1qmkln-1.
Therefore, ifk=1, then Theorem 6 implies(49)∑m∈P2amkqn1-qn1-qn+11-q2-n+1qndn-1qmkln-1=aqn1-qn1-qn+11-q2+n+1qndn-1qln-1tq-tq-3h2t.
which completes the proof of (46). The proof of (47) is similar and follows directly from the main theorem and Theorem 6 of Nelson.Theorem 17.
Fork=1, we have(50)∑m∈R2∑f∈Mnv1f=aqn1-qn1-qn+11-q2-n+1qndn-1qln-11tq-t2.(51)∑m∈R3∑f∈Mnv1f=n+1qndn-1qln-1-aqn1-qn1-qn+11-q21tq2-ttq-t2.Proof.
Using the main theorem, we obtain(52)∑m∈R2∑f∈Mnv1f=∑m∈R2amkqn1-qn1-qn+11-q2-n+1qndn-1qmkln-1.
Therefore, ifk=1, then Theorem 6 implies(53)∑m∈Mnamkqn1-qn1-qn+11-q2-n+1qndn-1qmkln-1=aqn1-qn1-qn+11-q2-n+1qndn-1qln-11tq-t2.
which complete the proof of (50). The proof of (51) is similar and follows directly from the main theorem and Theorem 6 of Nelson.Example 18.
In this example, we letm runs through the set of all linear polynomials.
Letn∈N and k∈N. If 1≤k≤q and u=(m-1)mk-1, then(54)∑m∈M1∑f∈Mn∑b∈M<n∑d∈Md∣fdbu+amk=aqq-1t-tqk.


## 4.1. Proof of the Main Theorem

Proof.
We havedbu+a=(ds2+s1)mk+(dr2+r1) and(35)dbu+amk=dbumk+amk.
Therefore, we have(36)vf=∑b∈M<n∑d∈Md∣fdbu+amk=∑b∈M<n∑d∈Md∣fdbumk+∑b∈M<n∑d∈Md∣famk=∑b∈M<n∑s2∈M∑d∈Md∣fdbu/mk=ds2ds2+∑b∈M<n∑s1∈M∑d∈Md∣fa/mk=s1s1
Rewriting the summation in terms ofr1 and r2 yields(37)vf=∑b∈M<n∑d∈Md∣fa=ms1+r1a-r1mk+∑b∈M<n∑r2∈M<n∑d∈Md∣fbu=ms2+r2dbu-r2mk.
Hence Lemmas10 and 13 imply(38)vf=1-m1-qmka-r1df+dn-1mk-1ln-1mk-1d0mkσfmk-dfmk∑b∈M<n∑r2∈M<nbm-1=ms2+r2r2=dn-1mk-1ln-1mk-1d0mkσfmk+df1-m1-qmka-r1-1mk∑b∈M<n∑r2∈M<nbm-1=ms2+r2r2.
Therefore,(39)∑f∈Mnvf=dn-1mk-1mkln-1mk-1d0mk∑f∈Mnσf-1-m1-qmka-r1-1mk∑b∈M<n∑r2∈M<nbu=mks2+r2r2∑f∈Mndf=dn-1mk-1mkln-1mk-1d0mkq2n·1-q-n-11-q-1-n+1qn1-m1-qmka-r1-1mk∑b∈M<n∑r2∈M<nbu=mks2+r2r2.
The proof of (18) is similar and is a direct consequence of Lemma 13.Corollary 14.
Ifm=p∈Pn, then(40)∑f∈Mnvf=dn-1pk-1ln-1pk-1d0pkpkq2n·1-q-n-11-q-1-n+1qn1-p1-qpka-r1-1pk1-p1-q.Proof.
If∈Pn, then, for each b∈M<n,bu=pks2+r2, we have (41)bu=bp-1pk-1≡r2mod⁡pk,which forces r2=1 by Euler theorem for polynomial; see [1]. Therefore,(42)∑b∈M<n∑r2∈M<nbp-1=ps2+r2r2=1-p1-q,and the corollary follows from the main theorem.


## 4.2. Evaluation of Some Sums inFq[t]

Theorem 15.
Letn∈N and for k∈N.
If1≤k≤q, then(43)∑m∈Mn∑f∈Mnv1f=aqn1-qn1-qn+11-q2-n+1qndn-1qln-11lnk.Proof.
Using the main theorem, we obtain(44)∑m∈Mn∑f∈Mnv1f=∑m∈Mnamkqn1-qn1-qn+11-q2-n+1qndn-1qmkln-1.
Therefore, if1≤k≤q, Theorem 5 implies(45)∑m∈Mnamkqn1-qn1-qn+11-q2-n+1qndn-1qmkln-1=aqn1-qn1-qn+11-q2-n+1qndn-1qln-11lnk.
which complete the proof of (43).Theorem 16.
Fork=1, we have (46)∑m∈P2∑f∈Mnv1f=atq-tq-3h2tqn1-qn1-qn+11-q2+n+1qndn-1qln-1tq-tq-3h2t.(47)∑m∈P3∑f∈Mnv1f=ah2t3tq2-tq-4h3tqn1-qn1-qn+11-q2-n+1qndn-1qln-1h2t3tq2-tq-4h3t.Proof.
By the main theorem, we have(48)∑m∈P2∑f∈Mnv1f=∑m∈P2amkqn1-qn1-qn+11-q2-n+1qndn-1qmkln-1.
Therefore, ifk=1, then Theorem 6 implies(49)∑m∈P2amkqn1-qn1-qn+11-q2-n+1qndn-1qmkln-1=aqn1-qn1-qn+11-q2+n+1qndn-1qln-1tq-tq-3h2t.
which completes the proof of (46). The proof of (47) is similar and follows directly from the main theorem and Theorem 6 of Nelson.Theorem 17.
Fork=1, we have(50)∑m∈R2∑f∈Mnv1f=aqn1-qn1-qn+11-q2-n+1qndn-1qln-11tq-t2.(51)∑m∈R3∑f∈Mnv1f=n+1qndn-1qln-1-aqn1-qn1-qn+11-q21tq2-ttq-t2.Proof.
Using the main theorem, we obtain(52)∑m∈R2∑f∈Mnv1f=∑m∈R2amkqn1-qn1-qn+11-q2-n+1qndn-1qmkln-1.
Therefore, ifk=1, then Theorem 6 implies(53)∑m∈Mnamkqn1-qn1-qn+11-q2-n+1qndn-1qmkln-1=aqn1-qn1-qn+11-q2-n+1qndn-1qln-11tq-t2.
which complete the proof of (50). The proof of (51) is similar and follows directly from the main theorem and Theorem 6 of Nelson.Example 18.
In this example, we letm runs through the set of all linear polynomials.
Letn∈N and k∈N. If 1≤k≤q and u=(m-1)mk-1, then(54)∑m∈M1∑f∈Mn∑b∈M<n∑d∈Md∣fdbu+amk=aqq-1t-tqk.


## 5. Conclusion

In this project, we prove the polynomial analogue of some Liouville theorems involving the arithmetic polynomial divisor functions. We then use the results obtained to establish several polynomials identities over the finite fields ofq elements. As an example, we show that some complex sums of polynomials in Fq[t] may be reduced to very simple algebraic expressions.



---

*Source: 2896847-2019-07-09.xml*