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Connecting two types of representations of a permutation of Fq
Discrete Mathematics 343 (2020) 111793
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Connecting two types of representations of a permutation of Fq ✩ Zhiguo Ding Zhongsheng School of Advanced Study, Hunan Institute of Traffic Engineering, Changsha, Hunan 410005, China
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Article history: Received 12 December 2017 Received in revised form 16 December 2019 Accepted 18 December 2019 Available online xxxx Keywords: Representation Permutation polynomial Finite field Rational function Projective line Carlitz rank
a b s t r a c t In this paper, we connect two types of representations of a permutation σ of the finite field Fq . One type is algebraic, in which the permutation is represented as the composition of degree-one polynomials and k copies of xq−2 , for some prescribed value of k. The other type is combinatorial, in which the permutation is represented as the composition of a degree-one rational function followed by the product of k 2-cycles on P1 (Fq ) := Fq ∪ {∞}, where each 2-cycle moves ∞. We show that, after modding out by obvious equivalences amongst the algebraic representations, then for each k there is a bijection between the algebraic representations of σ and the combinatorial representations of σ . We also prove analogous results for permutations of P1 (Fq ). One consequence is a new characterization of the notion of Carlitz rank of a permutation on Fq , which we use elsewhere to provide an explicit formula for the Carlitz rank. Another consequence involves a classical theorem of Carlitz, which says that if q > 2 then the group of permutations of Fq is generated by the permutations induced by degree-one polynomials and xq−2 . Our bijection provides a new perspective from which the two proofs of this result in the literature can be seen to arise naturally, without requiring the clever tricks that previously appeared to be needed in order to discover those proofs. © 2019 Elsevier B.V. All rights reserved.
1. Introduction Throughout, we assume that q > 2 is a prime power and Fq is the finite field with q elements. Note that permutations of Fq can be viewed as permutations of P1 (Fq ) := Fq ∪ {∞} which fixes ∞. We will study two completely different types of representations of permutations on P1 (Fq ). The first type of representations is essentially algebraical, and the second type of representations is essentially combinatorial. More precisely, fixing a positive integer k and a permutation σ of P1 (Fq ), we denote by Aσ ,k the set of all representations of σ of the form
σ = µ(x) ◦ xq−2 ◦ (x − ak ) ◦ xq−2 ◦ (x − ak−1 ) ◦ · · · ◦ xq−2 ◦ (x − a1 ) with a1 , a2 , . . . , ak ∈ Fq and µ(x) ∈ Fq (x) of degree one, and similarly we denote by Cσ ,k the set of all representations of σ of the form
σ = ν (x) ◦ (b1 , ∞) ◦ (b2 , ∞) ◦ · · · ◦ (bk , ∞) with b1 , b2 , . . . , bk ∈ Fq and ν (x) ∈ Fq (x) of degree one. ✩ A version of Theorem 1 of the present paper was originally proved as part of a joint project with Michael Zieve which resulted in the paper Ding and Zieve (2019). The author thanks Michael Zieve for encouraging him to publish Theorem 1 separately in the present paper. The author thanks the referees and Michael Zieve for suggesting improvements to the statement and proof of Theorem 1, which yielded a stronger result than appeared in the original version of this paper. E-mail address: [email protected]. https://doi.org/10.1016/j.disc.2019.111793 0012-365X/© 2019 Elsevier B.V. All rights reserved.
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The presentations in the set Aσ ,k are algebraically nice, since they are compositions of a degree-one rational function
µ(x) with the monomial xq−2 and monic degree-one polynomials x − ai with 1 ≤ i ≤ k. The presentations in the set Cσ ,k are combinatorially nice, since they are compositions of a degree-one rational function ν (x) with 2-cycles on P1 (Fq ) of the form (bi , ∞) with 1 ≤ i ≤ k.
Although Aσ ,k and Cσ ,k are completely different in nature, Theorem 7 gives a recipe to turn any representation in Aσ ,k into a representation in Cσ ,k , and Theorem 8 gives a recipe to turn any representation in Cσ ,k into a representation in Aσ ,k . Moreover, due to Theorem 6 the maps F : Aσ ,k → Cσ ,k and G : Cσ ,k → Aσ ,k induced by the above two recipes respectively are inverses to one another. In other words, there exist naturally two inverse bijections between Aσ ,k and Cσ ,k , which are induced by the recipes as illustrated in Theorems 7 and 8 respectively, as stated in the following: Theorem 1. For any positive integer k and any permutation σ of P1 (Fq ) := Fq ∪ {∞} with q > 2, there are two natural inverse bijections between the set Aσ ,k of all representations of σ of the form
σ = µ(x) ◦ xq−2 ◦ (x − ak ) ◦ xq−2 ◦ (x − ak−1 ) ◦ · · · ◦ xq−2 ◦ (x − a1 ) with a1 , a2 , . . . , ak ∈ Fq and µ(x) ∈ Fq (x) of degree one and the set Cσ ,k of all representations of σ of the form
σ = ν (x) ◦ (b1 , ∞) ◦ (b2 , ∞) ◦ · · · ◦ (bk , ∞) with b1 , b2 , . . . , bk ∈ Fq and ν (x) ∈ Fq (x) of degree one. In particular, the finite sets Aσ ,k and Cσ ,k have the same cardinality. There are two easy consequences of our results. The first consequence is a new characterization of the notion of Carlitz rank of a permutation on Fq , based on which an explicit formula about Carlitz rank has been obtained in [11]. The second consequence involves a classical theorem of Carlitz, which says that if q > 2 then the group of permutations of Fq is generated by the permutations induced by degree-one polynomials and xq−2 . Our bijection in Theorem 1 provides a new perspective from which the two proofs of this result in the literature can be seen to arise naturally, without requiring the clever tricks that previously appeared to be needed in order to discover those proofs. This paper is organized as follows. In Section 2, we prove some results about permutations on P1 (Fq ) which will be used in our treatment. In Section 3 we give a proof for Theorem 1 together with the explicit recipe, which turns an algebraically nice representation of a permutation of P1 (Fq ) into a combinatorially nice representation of the same permutation and vice versa. We conclude in Section 4 by illustrating the above mentioned two consequences of our results. 2. Basic facts Let us begin with the following basic observation: Lemma 2.
We have (0, ∞) = x−1 ◦ xq−2 as permutations of P1 (Fq ).
Proof. Note that xq−2 = x−1 for any nonzero element x ∈ Fq , so the map x−1 ◦ xq−2 fixes each element in F∗q . It is not hard to verify directly that x−1 ◦ xq−2 exchanges the remaining two points 0 and ∞. □ Next, we need to know the conjugations of a given 2-cycle (b, ∞) with b ∈ Fq by a degree-one polynomial x − a and by the monomial xq−2 respectively. For this purpose let us give first the following fact: Lemma 3. u, v ∈ X .
The relation f ◦ (u, v ) = (f (u), f (v )) ◦ f holds for any injective map f of sets from X to Y and any two distinct
Proof. Both f ◦ (u, v ) and (f (u), f (v )) ◦ f send x to f (x) for any element x ∈ X \ {u, v}, while both of them send u to f (v ) and send v to f (u). □ Corollary 4.
The following identities hold as permutations of P1 (Fq ):
• (x − a) ◦ (b, ∞) = (b − a, ∞) ◦ (x − a) for any a, b ∈ Fq , • xq−2 ◦ (b, ∞) = (bq−2 , ∞) ◦ xq−2 for any b ∈ Fq . Proof. The results follows from Lemma 3 by taking X = Y := P1 (Fq ), u := b, v := ∞, letting f (x) := x − a and f (x) := xq−2 respectively. □ Corollary 4 implies particularly that xq−2 ◦ (0, ∞) = (0, ∞) ◦ xq−2 , which is just equal to x−1 by Lemma 2. Moreover, it is interesting to note that the composition of any two of the three permutations x−1 , xq−2 , (0, ∞) of P1 (Fq ) is equal to the third one. In other words, the subgroup of permutations of P1 (Fq ) with q > 2 generated by x−1 , xq−2 , and (0, ∞) is exactly the Klein four-group. After conjugating (0, ∞) by the translation x + a, we can obtain the following expression of a 2-cycle of the form (a, ∞) with a ∈ Fq :
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Corollary 5.
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(a, ∞) = (x−1 + a) ◦ xq−2 ◦ (x − a) holds for any a ∈ Fq .
Proof. By applications of Corollary 4 and Lemma 2, we have (a, ∞) = (x + a) ◦ (0, ∞) ◦ (x − a) = (x + a) ◦ x−1 ◦ xq−2 ◦ (x − a), which is equal to (x−1 + a) ◦ xq−2 ◦ (x − a) obviously.
□
3. Proof of Theorem 1 For a fixed positive integer k, let us define maps F : Fkq → Fkq and G : Fkq → Fkq as follows, which will be used to illustrate the recipes in Theorems 7 and 8 respectively. Define F : Fkq → Fkq by sending (a1 , a2 , . . . , ak ) to (b1 , b2 , . . . , bk ), where bi := ci,i for 1 ≤ i ≤ k, and for any fixed q−2
1 ≤ i ≤ k we define ci,j with 0 ≤ j ≤ i inductively by ci,0 := 0 and ci,j := ci,j−1 + ai−j+1 . 1 In order to define the map G : Fkq → Fkq , we need to use the map Φℓ : Fℓq → Fℓ− for each ℓ with 2 ≤ ℓ ≤ k, which q is defined to send (e1 , e2 , . . . , eℓ ) to ((e2 − e1 )q−2 , (e3 − e1 )q−2 , . . . , (eℓ − e1 )q−2 ). For ease of notation, we will drop the subscript ℓ and write Φ for Φℓ when the value of ℓ is clear from context. Define G : Fkq → Fkq by sending (b1 , b2 , . . . , bk ) to (a1 , a2 , . . . , ak ), where for 1 ≤ i ≤ k we denote by ai the first entry of Φ i−1 (b1 , b2 , . . . , bk ), in which the map Φ i−1 : Fkq → Fqk−i+1 is really the composition Φk−i+2 ◦ Φk−i+3 ◦ · · · ◦ Φk with the convention that Φ 0 is the identity map on Fkq . More precisely, if we write
Φ i−1 (b1 , b2 , . . . , bk ) = (d1,i−1 , d2,i−1 , . . . , dk−i+1,i−1 ) for 1 ≤ i ≤ k, then by definition we have (a1 , a2 , . . . , ak ) := (d1,0 , d1,1 , . . . , d1,k−1 ). Theorem 6. on Fkq .
The maps F : Fkq → Fkq and G : Fkq → Fkq are inverses of one another. In particular, both F and G are bijections
Proof. It suffices to show that G ◦ F is the identity map on Fkq , since then F is injective and G is surjective, but since each of F and G is a map on Fkq , it follows that F and G are bijective, whence F and G are inverse bijections since G ◦ F is the identity map on Fkq again. Now, let us show that G ◦ F is the identity map on Fkq . Given a1 , a2 , . . . , ak in Fq , recall that for 1 ≤ i ≤ k the elements q−2
ci,j with 0 ≤ j ≤ i are defined inductively by ci,0 = 0 and ci,j = ci,j−1 + ai−j+1 . Put bj = cj,j for 1 ≤ j ≤ k, so that by the definition of F we get F (a1 , a2 , . . . , ak ) = (b1 , b2 , . . . , bk ). For 1 ≤ i ≤ k let us write
Φ i−1 (b1 , b2 , . . . , bk ) = (d1,i−1 , d2,i−1 , . . . , dk−i+1,i−1 ), then by the definition of G we have G(b1 , b2 , . . . , bk ) = (d1,0 , d1,1 , . . . , d1,k−1 ). We will show by induction on j that di,j = ci+j,i holds for all i, j with 0 ≤ j ≤ k − 1 and 1 ≤ i ≤ k − j. The base case j = 0 says that di,0 = ci,i for 1 ≤ i ≤ k, which is true since both sides equal bi . Inductively, if 1 ≤ j ≤ k − 1, then for each 1 ≤ i ≤ k − j we have di,j = (di+1,j−1 − d1,j−1 )q−2 = (ci+j,i+1 − cj,1 )q−2
=(ci+j,i+1 − aj )q−2 = (ciq+−j,2i )q−2 = ci+j,i , which concludes the induction. Thus G ◦ F (a1 , a2 , . . . , ak ) = G(b1 , b2 , . . . , bk )
=(d1,0 , d1,1 , . . . , d1,k−1 ) = (c1,1 , c2,1 , . . . , ck,1 ) = (a1 , a2 , . . . , ak ). So G ◦ F is the identity map on Fkq , whence as explained above it follows that F and G are inverse bijections on Fkq . Theorem 7.
For any a1 , a2 , . . . , ak in Fq , denote
(b1 , b2 , . . . , bk ) := F (a1 , a2 , . . . , ak ),
□
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then xq−2 ◦ (x − ak ) ◦ xq−2 ◦ (x − ak−1 ) ◦ · · · ◦ xq−2 ◦ (x − a1 )
=ν (x) ◦ (b1 , ∞) ◦ (b2 , ∞) ◦ · · · ◦ (bk , ∞), where
ν (x) := x−1 ◦ (x − ak ) ◦ x−1 ◦ (x − ak−1 ) ◦ · · · ◦ x−1 ◦ (x − a1 ) is a degree-one rational function in Fq (x). Proof. We prove it by induction on k. The base case k = 1 says xq−2 ◦ (x − a1 ) = x−1 ◦ (x − a1 ) ◦ (b1 , ∞) where b1 = F (a1 ) = a1 . This identity follows directly from Corollary 5. Inductively, suppose k > 1 and a1 , a2 , . . . , ak are q−2 in Fq . By definition, for 1 ≤ i ≤ k we have ci,0 = 0 and ci,j = ci,j−1 + ai−j+1 with 1 ≤ j ≤ i. Write bi = ci,i for 1 ≤ i ≤ k, so that (b1 , b2 , . . . , bk ) = F (a1 , a2 , . . . , ak ). For ease of expression, let us denote
λi := x−1 ◦ (x − ak ) ◦ xq−2 ◦ (x − ak−1 ) ◦ xq−2 ◦ · · · ◦ xq−2 ◦ (x − ai ) for 1 ≤ i ≤ k, and denote
ρi := xq−2 ◦ (x − ai ) ◦ xq−2 ◦ (x − ai−1 ) ◦ · · · ◦ xq−2 ◦ (x − a1 ) for 0 ≤ i ≤ k where by convention ρ0 = x. Write
σ := xq−2 ◦ (x − ak ) ◦ xq−2 ◦ (x − ak−1 ) ◦ · · · ◦ xq−2 ◦ (x − a1 ). Hence we have
σ = ρk =xq−2 ◦ (x − ak ) ◦ ρk−1 =x−1 ◦ (0, ∞) ◦ (x − ak ) ◦ ρk−1 =x−1 ◦ (x − ak ) ◦ (ak , ∞) ◦ ρk−1 =λk ◦ (ck,1 , ∞) ◦ ρk−1 . We claim that
λi+1 ◦ (ck,k−i , ∞) ◦ ρi = λi ◦ (ck,k−i+1 , ∞) ◦ ρi−1 for 1 ≤ i ≤ k − 1. Indeed, for each i with 1 ≤ i ≤ k − 1, we have (ck,k−i , ∞) ◦ ρi
=(ck,k−i , ∞) ◦ xq−2 ◦ (x − ai ) ◦ ρi−1 =xq−2 ◦ (ckq,−k−2 i , ∞) ◦ (x − ai ) ◦ ρi−1 =xq−2 ◦ (x − ai ) ◦ (ck,k−i+1 , ∞) ◦ ρi−1 . Thus by prepending λi+1 to these permutations we get
λi+1 ◦ (ck,k−i , ∞) ◦ ρi =λi+1 ◦ xq−2 ◦ (x − ai ) ◦ (ck,k−i+1 , ∞) ◦ ρi−1 =λi ◦ (ck,k−i+1 , ∞) ◦ ρi−1 , which concludes the proof of the claim. So we have
σ = λk ◦ (ck,1 , ∞) ◦ ρk−1 = λk−1 ◦ (ck,2 , ∞) ◦ ρk−2 = · · · = λ1 ◦ (ck,k , ∞) ◦ ρ0 , which by definition is equal to x−1 ◦ (x − ak ) ◦ xq−2 ◦ (x − ak−1 ) ◦ xq−2 ◦ · · · ◦ xq−2 ◦ (x − a1 ) ◦ (bk , ∞). By inductive hypothesis, we have xq−2 ◦ (x − ak−1 ) ◦ xq−2 ◦ (x − ak−2 ) ◦ · · · ◦ xq−2 ◦ (x − a1 )
=x−1 ◦ (x − ak−1 ) ◦ x−1 ◦ (x − ak−2 ) ◦ · · · ◦ x−1 ◦ (x − a1 )◦ ◦ (b1 , ∞) ◦ (b2 , ∞) ◦ · · · ◦ (bk−1 , ∞).
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Therefore, we get
σ =x−1 ◦ (x − ak ) ◦ x−1 ◦ (x − ak−1 ) ◦ · · · ◦ x−1 ◦ (x − a1 )◦ ◦ (b1 , ∞) ◦ (b2 , ∞) ◦ · · · ◦ (bk , ∞), which concludes the induction and hence concludes the proof. Theorem 8.
□
For any b1 , b2 , . . . , bk in Fq , denote
(a1 , a2 , . . . , ak ) := G(b1 , b2 , . . . , bk ), then (b1 , ∞) ◦ (b2 , ∞) ◦ · · · ◦ (bk , ∞)
=µ(x) ◦ xq−2 ◦ (x − ak ) ◦ xq−2 ◦ (x − ak−1 ) ◦ · · · ◦ xq−2 ◦ (x − a1 ), where
µ(x) := (x + a1 ) ◦ x−1 ◦ (x + a2 ) ◦ x−1 ◦ · · · ◦ (x + ak ) ◦ x−1 is a degree-one rational function in Fq (x). Proof. Let us prove it by induction on k. The base case k = 1 says (b1 , ∞) = (x + a1 ) ◦ x−1 ◦ xq−2 ◦ (x − a1 ) where a1 = G(b1 ) = b1 , which is true by Corollary 5. For the inductive step, assume that k > 1 and b1 , b2 , . . . , bk are in Fq . Write
Φ i−1 (b1 , b2 , . . . , bk ) = (d1,i−1 , d2,i−1 , . . . , dk−i+1,i−1 ) for 1 ≤ i ≤ k, so that (a1 , a2 , . . . , ak ) = G(b1 , b2 , . . . , bk ) = (d1,0 , d1,1 , . . . , d1,k−1 ). For ease of expression, let us define
γi := (x + b1 ) ◦ x−1 ◦ (d1,1 , ∞) ◦ (d2,1 , ∞) ◦ · · · ◦ (di−1,1 , ∞) ◦ xq−2 ◦ (x − b1 ) for 1 ≤ i ≤ k, and define
θi := (bi+1 , ∞) ◦ (bi+2 , ∞) ◦ . . . ◦ (bk , ∞) for 0 ≤ i ≤ k, where by convention γ1 = (x + b1 ) ◦ x−1 ◦ xq−2 ◦ (x − b1 ) and θk = x. Denote
τ := (b1 , ∞) ◦ (b2 , ∞) ◦ · · · ◦ (bk , ∞). Hence by Corollary 5 we have
τ = θ0 = (b1 , ∞) ◦ θ1 = (x + b1 ) ◦ x−1 ◦ xq−2 ◦ (x − b1 ) ◦ θ1 = γ1 ◦ θ1 . We claim that γi ◦ θi = γi+1 ◦ θi+1 for 1 ≤ i ≤ k − 1. Indeed, for each i with 1 ≤ i ≤ k − 1, we have xq−2 ◦ (x − b1 ) ◦ θi
=xq−2 ◦ (x − b1 ) ◦ (bi+1 , ∞) ◦ θi+1 =xq−2 ◦ (bi+1 − b1 , ∞) ◦ (x − b1 ) ◦ θi+1 =(di,1 , ∞) ◦ xq−2 ◦ (x − b1 ) ◦ θi+1 . Thus by prepending (x + b1 ) ◦ x−1 ◦ (d1,1 , ∞) ◦ (d2,1 , ∞) ◦ · · · ◦ (di−1,1 , ∞) to these permutations, we get
γi ◦ θi =(x + b1 ) ◦ x−1 ◦ (d1,1 , ∞) ◦ (d2,1 , ∞) ◦ · · · ◦ (di−1,1 , ∞) ◦ xq−2 ◦ (x − b1 ) ◦ θi =(x + b1 ) ◦ x−1 ◦ (d1,1 , ∞) ◦ (d2,1 , ∞) ◦ · · · ◦ (di−1,1 , ∞) ◦ (di,1 , ∞) ◦ xq−2 ◦ (x − b1 ) ◦ θi+1 =γi+1 ◦ θi+1 ,
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which concludes the proof of the claim. So we have
τ = γ1 ◦ θ1 = γ2 ◦ θ2 = · · · = γk ◦ θk which by definition is equal to (x + b1 ) ◦ x−1 ◦ (d1,1 , ∞) ◦ (d2,1 , ∞) ◦ · · · ◦ (dk−1,1 , ∞) ◦ xq−2 ◦ (x − b1 ). By inductive hypothesis, if we denote (aˆ1 , aˆ2 , . . . , aˆ k−1 ) := G(d1,1 , d2,1 , . . . , dk−1,1 ), then (d1,1 , ∞) ◦ (d2,1 , ∞) ◦ · · · ◦ (dk−1,1 , ∞) −1 =(x + aˆ1 ) ◦ x−1 ◦ (x + aˆ2 ) ◦ x−1 ◦ · · · ◦ (x + aˆ ◦ k−1 ) ◦ x
q−2 q−2 ◦ xq−2 ◦ (x − aˆ ◦ (x − aˆ ◦ (x − aˆ1 ). k−1 ) ◦ x k−2 ) ◦ · · · ◦ x
Note that for 1 ≤ i ≤ k − 1 we have ˆ ai = ai+1 since
Φ i−1 (d1,1 , d2,1 , . . . , dk−1,1 ) = Φ i (b1 , b2 , . . . , bk ). Therefore, we have
τ =(x + b1 ) ◦ x−1 ◦ (x + a2 ) ◦ x−1 ◦ (x + a3 ) ◦ x−1 ◦ · · · ◦ (x + ak ) ◦ x−1 ◦ xq−2 ◦ (x − ak ) ◦ xq−2 ◦ (x − ak−1 ) ◦ · · · ◦ xq−2 ◦ (x − a2 ) ◦ xq−2 ◦ (x − b1 ), which concludes the induction and the proof since b1 = a1 .
□
We remark that Theorem 7 gives a recipe for turning an algebraical presentation of a permutation on P1 (Fq ) of the form
µ(x) ◦ xq−2 ◦ (x − ak ) ◦ xq−2 ◦ (x − ak−1 ) ◦ · · · ◦ xq−2 ◦ (x − a1 ) into a combinatorial presentation of the same permutation of the form
ν (x) ◦ (b1 , ∞) ◦ (b2 , ∞) ◦ · · · ◦ (bk , ∞), and conversely Theorem 8 gives a recipe for turning a combinatorial presentation of a permutation on P1 (Fq ) of the form
ν (x) ◦ (b1 , ∞) ◦ (b2 , ∞) ◦ · · · ◦ (bk , ∞) into an algebraical presentation of the same permutation of the form
µ(x) ◦ xq−2 ◦ (x − ak ) ◦ xq−2 ◦ (x − ak−1 ) ◦ · · · ◦ xq−2 ◦ (x − a1 ), where a1 , a2 . . . , ak , b1 , b2 . . . , bk are elements in Fq and µ(x), ν (x) are degree-one rational functions in Fq (x). In other words, for any fixed permutation σ of P1 (Fq ), there are two natural maps F : Aσ ,k → Cσ ,k and G : Cσ ,k → Aσ ,k induced by the recipes illustrated in Theorems 7 and 8 respectively, where Aσ ,k is the set of all representations of σ of the form
σ = µ(x) ◦ xq−2 ◦ (x − ak ) ◦ xq−2 ◦ (x − ak−1 ) ◦ · · · ◦ xq−2 ◦ (x − a1 ) with a1 , a2 , . . . , ak ∈ Fq and µ(x) ∈ Fq (x) of degree one, and Cσ ,k is the set of all representations of σ of the form
σ = ν (x) ◦ (b1 , ∞) ◦ (b2 , ∞) ◦ · · · ◦ (bk , ∞) with b1 , b2 , . . . , bk ∈ Fq and ν (x) ∈ Fq (x) of degree one. Now we are ready to prove Theorem 1, which asserts the natural maps F : Aσ ,k → Cσ ,k and G : Cσ ,k → Aσ ,k induced by the recipes in Theorems 7 and 8 respectively are inverses to one another. Proof of Theorem 1. It suffices to show that both G ◦ F and F ◦ G are the identity maps, where F : Aσ ,k → Cσ ,k and G : Cσ ,k → Aσ ,k are the natural maps induced by Theorems 7 and 8 respectively. Given any algebraic representation of σ in the set Aσ ,k of the form
σ = µ(x) ◦ xq−2 ◦ (x − ak ) ◦ xq−2 ◦ (x − ak−1 ) ◦ · · · ◦ xq−2 ◦ (x − a1 ) with a1 , a2 , . . . , ak ∈ Fq and µ(x) ∈ Fq (x) of degree one, by definition F sends it to a combinatorial representation of σ in Cσ ,k of the form
σ = ν (x) ◦ (b1 , ∞) ◦ (b2 , ∞) ◦ · · · ◦ (bk , ∞)
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with b1 , b2 , . . . , bk ∈ Fq and ν (x) ∈ Fq (x) of degree one, in which (b1 , b2 , . . . , bk ) := F (a1 , a2 , . . . , ak ),
ν0 (x) := x−1 ◦ (x − ak ) ◦ x−1 ◦ (x − ak−1 ) ◦ · · · ◦ x−1 ◦ (x − a1 ), and ν (x) := µ(x) ◦ ν0 (x). This combinatorial representation is sent by G to another algebraic representation of σ in Aσ ,k of the form
σ = µ∗ (x) ◦ xq−2 ◦ (x − a∗k ) ◦ xq−2 ◦ (x − a∗k−1 ) ◦ · · · ◦ xq−2 ◦ (x − a∗1 ) with a∗1 , a∗2 , . . . , a∗k ∈ Fq and µ∗ (x) ∈ Fq (x) of degree one, in which (a∗1 , a∗2 , . . . , a∗k ) := G(b1 , b2 , . . . , bk ),
µ0 (x) := (x + a∗1 ) ◦ x−1 ◦ (x + a∗2 ) ◦ x−1 ◦ · · · ◦ (x + a∗k ) ◦ x−1 , and µ∗ (x) := ν (x) ◦ µ0 (x). By Theorem 6, the k-tuple (a∗1 , a∗2 , . . . , a∗k ) = G(b1 , b2 , . . . , bk ) = G ◦ F (a1 , a2 , . . . , ak ) equals (a1 , a2 , . . . , ak ), i.e., we have a∗i = ai for any 1 ≤ i ≤ k. Thus µ0 (x) and ν0 (x) are inverse to one another, which implies that
µ∗ (x) = ν (x) ◦ µ0 (x) = µ(x) ◦ ν0 (x) ◦ µ0 (x) = µ(x). Hence, G ◦ F is the identity map on the set Aσ ,k . Similarly, we can show that F ◦ G is the identity map on Cσ ,k . Therefore, the natural maps F : Aσ ,k → Cσ ,k and G : Cσ ,k → Aσ ,k induced by Theorems 7 and 8 respectively are inverse bijections. □ 4. Consequences Note that any permutation of Fq can be extended uniquely to a permutation of P1 (Fq ). Hence permutations of Fq can be regarded as permutations of P1 (Fq ) in this sense. By application of Theorem 1 to permutations of Fq , we get the following result: Corollary 9. Suppose f is a permutation of Fq , and then extends f to a permutation of P1 (Fq ) which fixes ∞. Then in any representation f = µ(x) ◦ xq−2 ◦ (x − ak ) ◦ xq−2 ◦ (x − ak−1 ) ◦ · · · ◦ xq−2 ◦ (x − a1 ) with a1 , a2 , . . . , ak ∈ Fq and µ(x) ∈ Fq (x) of degree one, the rational function µ(x) must be a degree-one polynomial in Fq [x]. Therefore, the set of all representations of f in the form of
µ(x) ◦ xq−2 ◦ (x − ak ) ◦ xq−2 ◦ (x − ak−1 ) ◦ · · · ◦ xq−2 ◦ (x − a1 ) with a1 , a2 , . . . , ak ∈ Fq and µ(x) ∈ Fq [x] of degree one is naturally bijective to the set of all representations of f in the form of
ν (x) ◦ (b1 , ∞) ◦ (b2 , ∞) ◦ · · · ◦ (bk , ∞) with b1 , b2 , . . . , bk ∈ Fq and ν (x) ∈ Fq (x) of degree one. Proof. Note that the degree-one rational function µ(x) ∈ Fq (x) in any representation of the extended permutation f on P1 (Fq ) in the form of f = µ(x) ◦ xq−2 ◦ (x − ak ) ◦ xq−2 ◦ (x − ak−1 ) ◦ · · · ◦ xq−2 ◦ (x − a1 ) fixes the point ∞, since all of f , xq−2 , and (x − ai ) fix ∞. Hence the first assertion follows from the fact that the degree-one rational functions which fix ∞ are precisely the degree-one polynomials. The second assertion is then a direct consequence of Theorem 1. □ The following is a classical result on permutations of Fq , which was posed as a question by Straus, and was first proved by Carlitz [3] and then proved again by Zieve [22]: Theorem 10. If q > 2 is a prime power, then every permutation of Fq is a composition of xq−2 and degree-one polynomials over Fq . Inspired by Theorem 10, Aksoy et al. [1] introduced the notion of the Carlitz rank of a permutation f of Fq , which means the smallest value of n ≥ 0 in any representation of f of the form f = θ0 ◦ xq−2 ◦ θ1 ◦ xq−2 ◦ . . . ◦ xq−2 ◦ θn
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in which each θi is a degree-one polynomial in Fq [x]. For a systematic study of the general theory of Carlitz ranks and their generalizations, see [11]. Carlitz ranks have been extensively studied in the literature, see for example the papers [1,6–9,14–17,19,20]. For the history of permutation polynomials over finite fields, see the papers [2–5,10,12,13,18,21]. As the first consequence, our results provide a new perspective on the theory of Carlitz ranks from the point of view of combinatorics, instead of from the point of view of algebra as in the literature previously. More precisely, by Corollary 9 the Carlitz rank of a permutation f of Fq is exactly the smallest integer n ≥ 0 for which the uniquely extended permutation of P1 (Fq ), which is also denoted by f by abuse of language, admits a combinatorial representation of the form f = ν (x) ◦ (b1 , ∞) ◦ (b2 , ∞) ◦ · · · ◦ (bn , ∞) for some b1 , b2 , . . . , bn ∈ Fq and some ν (x) ∈ Fq (x) of degree one. Based on this new characterization of Carlitz rank, the following result on the computation of Carlitz rank has be obtained in [11]: Theorem 11. Let f = µ ◦ σ be a permutation of Fq , where µ is a degree-one rational function, and σ is a permutation of P1 (Fq ) which moves s points in Fq and has t nontrivial orbits. Define n := s + t if σ (∞) = ∞ and n := s + t − 1 if σ (∞) ̸ = ∞. Then the Carlitz rank of f is at most n, and it equals n if in addition q ≥ n + s + 2. As the second consequence, our results provide new perspective to Theorem 10. Indeed, Theorem 10 follows easily from Corollary 9 in various ways, since any permutation f of Fq has many different ways to be represented as f = (b1 , ∞) ◦ (b2 , ∞) ◦ · · · ◦ (bk , ∞) for some k ≥ 1 and some b1 , b2 , . . . , bk ∈ Fq , and by Corollary 9 any such combinatorial representation of f gives rise to an algebraic representation of f as a composition of k copies of xq−2 and degree-one polynomials over Fq , which concludes our proof for Theorem 10. More precisely, given any nontrivial permutation f of Fq , we can write f as the product of disjoint cycles of length at least two as follows: f = (b11 , b12 , . . . , b1s1 )(b21 , b22 , . . . , b2s2 ) · · · (bt1 , bt2 , . . . , btst ), where t ≥ 1 and each si ≥ 2 with 1 ≤ i ≤ t. Note that (bi1 , bi2 , . . . , bisi ) = (bi1 , ∞) ◦ [(bisi , ∞) ◦ · · · ◦ (bi2 , ∞) ◦ (bi1 , ∞)], for any 1 ≤ i ≤ t. Hence we get a representation of f of the form f = (b1 , ∞) ◦ (b2 , ∞) ◦ · · · ◦ (bn , ∞), where s := Σit=1 si , k := s + t, and denote by (b1 , b2 , . . . , bk ) the tuple ((b11 , b1s1 , . . . , b11 ), (b21 , b2s2 , . . . , b21 ), . . . , (bt1 , btst , . . . , bt1 )). By Theorem 8, if we denote (a1 , a2 , . . . , ak ) := G(b1 , b2 , . . . , bk ), then we obtain a representation of f of the form f = µ(x) ◦ xq−2 ◦ (x − ak ) ◦ xq−2 ◦ (x − ak−1 ) ◦ · · · ◦ xq−2 ◦ (x − a1 ), where
µ(x) := (x + a1 ) ◦ x−1 ◦ (x + a2 ) ◦ x−1 ◦ · · · ◦ (x + ak ) ◦ x−1 ∈ Fq (x) is a rational function of degree one. Furthermore, µ(x) is a degree-one polynomial over Fq by Corollary 9. Therefore, we obtain an algebraic representation of the given permutation f of Fq as a composition of k copies of xq−2 and k + 1 degree-one polynomials over Fq as above. Moreover, if in addition q ≥ 2s + t + 2, then this representation of f is optimal in the sense that it has the least possible copies of xq−2 , since in this case the Carlitz rank of f is exactly k by Theorem 11. In particular, by application of the above procedure to 2-cycles of the form (0, a) with a ∈ F∗q , we can recover uniformly both proofs for Theorem 10 in Carlitz [3] and Zieve [22], which involve some clever tricks and have not been related to each other previously. First, let us review briefly the proofs for Theorem 10 in Carlitz [3] and Zieve [22]. The starting point for both proofs is the same, i.e., it is enough to show the result in the special case that the permutation is a 2-cycle of the form (0, a) with a ∈ F∗q , since any permutation of Fq can be written as a product of such 2-cycles. But Carlitz [3] and Zieve [22] give different expressions of the 2-cycle (0, a) as compositions of xq−2 and degree-one polynomials. More precisely, Carlitz [3] relies on the mysterious observation that (0, a) = (−a2 x) ◦ xq−2 ◦ (x − a) ◦ xq−2 ◦ (x + 1/a) ◦ xq−2 ◦ (x − a),
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but does not explain how it could be found; Zieve [22] observes that (0, a) = (−ax + a) ◦ xq−2 ◦ (−x + 1) ◦ xq−2 ◦ (−x + 1) ◦ xq−2 ◦ (x/a), which follows from the clever combination of Lemma 2 and the fact that the degree-one rational function 1 − x−1 induces an order-three permutation of P1 (Fq ) with a 3-cycle (∞, 1, 0). Now, we apply our procedure illustrated above to 2-cycles (0, a) with a ∈ F∗q to recover uniformly both of the above two observations in Carlitz [3] and Zieve [22] respectively. Indeed, for any a ∈ F∗q , there are two equally natural ways to express the 2-cycle (0, a) as a product of some 2-cycles of the form (b, ∞) with b ∈ Fq . More precisely, by our procedure the 2-cycle (0, a) can be expressed naturally as either (0, a) = (a, ∞) ◦ (0, ∞) ◦ (a, ∞) or (0, a) = (0, ∞) ◦ (a, ∞) ◦ (0, ∞). By some computation as explained above, the first identity leads exactly to the mysterious observation in Carlitz [3], and the second one gives (0, a) = (−a2 x + a) ◦ xq−2 ◦ (x + a) ◦ xq−2 ◦ (x − 1/a) ◦ xq−2 ◦ x, which is the same as the observation in Zieve [22] as polynomials. To see this it is enough to normalize all except the first degree-one polynomials to be monic in the observation discovered by Zieve [22]. Finally, we emphasize that it is of essential importance to introduce the extra point ∞ and to work over P1 (Fq ) instead of Fq , although both Carlitz’s original Theorem 10 and the notion of Carlitz rank are only about permutations of Fq and appear to have nothing to do with ∞. It is clear that neither of the above two consequences can be possibly obtained by working over Fq without introducing ∞. Declaration of competing interest The author declares that he has no conflicts of interest to this work. References [1] E. Aksoy, A. Çeşmelioğlu, W. Meidl, A. Topuzoğlu, On the Carlitz rank of permutation polynomials, Finite Fields Appl. 15 (2009) 428–440. [2] E. Betti, Sopra la risolubilità per radicali delle equazioni algebriche irriduttibili di grado primo, Ann. Sci. Mat. Fisiche 2 (1851) 5–19, (=Opere Matematiche, 1, 17–27). [3] L. Carlitz, Permutations in a finite field, Proc. Amer. Math. Soc. 4 (1953) 538. [4] L. Carlitz, A note on permutations in an arbitray field, Proc. Amer. Math. Soc. 14 (1963) 101. [5] L. Carlitz, Permutations in finite fields, Acta Sci. Math. (Szeged) 24 (1963) 196–203. [6] A. Çeşmelioğlu, A representation of permutations with full cycle, arXiv:1005.2019v1. [7] A. Çeşmelioğlu, W. Meidl, A. Topuzoğlu, Enumeration of a class of sequences generated by inversions, in: Coding and Cryptology, World Sci. Publ., Hackensack, NJ, 2008, pp. 44–57. [8] A. Çeşmelioğlu, W. Meidl, A. Topuzoğlu, On the cycle structure of permutation polynomials, Finite Fields Appl. 14 (2008) 593–614. [9] A. Çeşmelioğlu, W. Meidl, A. Topuzoğlu, Permutations of finite fields with prescribed properties, J. Comput. Appl. Math. 259 (part B) (2014) 536–545. [10] L.E. Dickson, The analytic representation of substitutions on a power of a prime number of letters, with a discussion of the linear group, Ann. Math. 11 (1896–1897) 65–120, and 161–183. [11] Z. Ding, M.E. Zieve, Carlitz ranks and approximate rational functions, 2019, in preparation. [12] K.D. Fryer, A class of permutation groups of prime degree, Canad. J. Math. 7 (1955) 24–34. [13] K.D. Fryer, Note on permutations in a finite field, Proc. Amer. Math. Soc. 6 (1955) 1–2. [14] D. Gomez-Perez, A. Ostafe, A. Topuzoğlu, On the Carlitz rank of permutations of Fq and pseudorandom sequences, J. Complexity 30 (2014) 279–289. [15] L. Işik, On complete mappings and value sets of polynomials over finite fields (Ph.D. thesis), Sabanci University, 2015. [16] L. Işik, A. Topuzoğlu, A note on value sets of polynomials over finite fields, 2017, arXiv:1701.06158v1. [17] L. Işik, A. Topuzoğlu, A. Winterhof, Complete mappings and Carlitz rank, Des. Codes Cryptogr. 85 (2017) 121–128. [18] R. Lidl, H. Niederreiter, Finite Fields, second ed., in: Encyclopedia Math. Appl., vol. 20, Cambridge Univ. Press, New York, 1997. [19] W. Meidl, A. Topuzoğlu, On the inversive pseudorandom number generator, in: Recent Developments in Applied Probability and Statistics, Physica, Heidelberg, 2010, pp. 103–125. [20] A. Topuzoğlu, The Carlitz rank of permutations of finite fields: A survey, J. Symbolic Comput. 64 (2014) 53–66. [21] C. Wells, Generators for groups of permutation polynomials over finite fields, Acta Sci. Math. Szeged. 29 (1968) 167–176. [22] M.E. Zieve, On a theorem of Carlitz, J. Group Theory 17 (2014) 667–669.
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Connecting two types of representations of a permutation of Fq ✩ Zhiguo Ding Zhongsheng School of Advanced Study, Hunan Institute of Traffic Engineering, Changsha, Hunan 410005, China
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Article history: Received 12 December 2017 Received in revised form 16 December 2019 Accepted 18 December 2019 Available online xxxx Keywords: Representation Permutation polynomial Finite field Rational function Projective line Carlitz rank
a b s t r a c t In this paper, we connect two types of representations of a permutation σ of the finite field Fq . One type is algebraic, in which the permutation is represented as the composition of degree-one polynomials and k copies of xq−2 , for some prescribed value of k. The other type is combinatorial, in which the permutation is represented as the composition of a degree-one rational function followed by the product of k 2-cycles on P1 (Fq ) := Fq ∪ {∞}, where each 2-cycle moves ∞. We show that, after modding out by obvious equivalences amongst the algebraic representations, then for each k there is a bijection between the algebraic representations of σ and the combinatorial representations of σ . We also prove analogous results for permutations of P1 (Fq ). One consequence is a new characterization of the notion of Carlitz rank of a permutation on Fq , which we use elsewhere to provide an explicit formula for the Carlitz rank. Another consequence involves a classical theorem of Carlitz, which says that if q > 2 then the group of permutations of Fq is generated by the permutations induced by degree-one polynomials and xq−2 . Our bijection provides a new perspective from which the two proofs of this result in the literature can be seen to arise naturally, without requiring the clever tricks that previously appeared to be needed in order to discover those proofs. © 2019 Elsevier B.V. All rights reserved.
1. Introduction Throughout, we assume that q > 2 is a prime power and Fq is the finite field with q elements. Note that permutations of Fq can be viewed as permutations of P1 (Fq ) := Fq ∪ {∞} which fixes ∞. We will study two completely different types of representations of permutations on P1 (Fq ). The first type of representations is essentially algebraical, and the second type of representations is essentially combinatorial. More precisely, fixing a positive integer k and a permutation σ of P1 (Fq ), we denote by Aσ ,k the set of all representations of σ of the form
σ = µ(x) ◦ xq−2 ◦ (x − ak ) ◦ xq−2 ◦ (x − ak−1 ) ◦ · · · ◦ xq−2 ◦ (x − a1 ) with a1 , a2 , . . . , ak ∈ Fq and µ(x) ∈ Fq (x) of degree one, and similarly we denote by Cσ ,k the set of all representations of σ of the form
σ = ν (x) ◦ (b1 , ∞) ◦ (b2 , ∞) ◦ · · · ◦ (bk , ∞) with b1 , b2 , . . . , bk ∈ Fq and ν (x) ∈ Fq (x) of degree one. ✩ A version of Theorem 1 of the present paper was originally proved as part of a joint project with Michael Zieve which resulted in the paper Ding and Zieve (2019). The author thanks Michael Zieve for encouraging him to publish Theorem 1 separately in the present paper. The author thanks the referees and Michael Zieve for suggesting improvements to the statement and proof of Theorem 1, which yielded a stronger result than appeared in the original version of this paper. E-mail address: [email protected]. https://doi.org/10.1016/j.disc.2019.111793 0012-365X/© 2019 Elsevier B.V. All rights reserved.
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The presentations in the set Aσ ,k are algebraically nice, since they are compositions of a degree-one rational function
µ(x) with the monomial xq−2 and monic degree-one polynomials x − ai with 1 ≤ i ≤ k. The presentations in the set Cσ ,k are combinatorially nice, since they are compositions of a degree-one rational function ν (x) with 2-cycles on P1 (Fq ) of the form (bi , ∞) with 1 ≤ i ≤ k.
Although Aσ ,k and Cσ ,k are completely different in nature, Theorem 7 gives a recipe to turn any representation in Aσ ,k into a representation in Cσ ,k , and Theorem 8 gives a recipe to turn any representation in Cσ ,k into a representation in Aσ ,k . Moreover, due to Theorem 6 the maps F : Aσ ,k → Cσ ,k and G : Cσ ,k → Aσ ,k induced by the above two recipes respectively are inverses to one another. In other words, there exist naturally two inverse bijections between Aσ ,k and Cσ ,k , which are induced by the recipes as illustrated in Theorems 7 and 8 respectively, as stated in the following: Theorem 1. For any positive integer k and any permutation σ of P1 (Fq ) := Fq ∪ {∞} with q > 2, there are two natural inverse bijections between the set Aσ ,k of all representations of σ of the form
σ = µ(x) ◦ xq−2 ◦ (x − ak ) ◦ xq−2 ◦ (x − ak−1 ) ◦ · · · ◦ xq−2 ◦ (x − a1 ) with a1 , a2 , . . . , ak ∈ Fq and µ(x) ∈ Fq (x) of degree one and the set Cσ ,k of all representations of σ of the form
σ = ν (x) ◦ (b1 , ∞) ◦ (b2 , ∞) ◦ · · · ◦ (bk , ∞) with b1 , b2 , . . . , bk ∈ Fq and ν (x) ∈ Fq (x) of degree one. In particular, the finite sets Aσ ,k and Cσ ,k have the same cardinality. There are two easy consequences of our results. The first consequence is a new characterization of the notion of Carlitz rank of a permutation on Fq , based on which an explicit formula about Carlitz rank has been obtained in [11]. The second consequence involves a classical theorem of Carlitz, which says that if q > 2 then the group of permutations of Fq is generated by the permutations induced by degree-one polynomials and xq−2 . Our bijection in Theorem 1 provides a new perspective from which the two proofs of this result in the literature can be seen to arise naturally, without requiring the clever tricks that previously appeared to be needed in order to discover those proofs. This paper is organized as follows. In Section 2, we prove some results about permutations on P1 (Fq ) which will be used in our treatment. In Section 3 we give a proof for Theorem 1 together with the explicit recipe, which turns an algebraically nice representation of a permutation of P1 (Fq ) into a combinatorially nice representation of the same permutation and vice versa. We conclude in Section 4 by illustrating the above mentioned two consequences of our results. 2. Basic facts Let us begin with the following basic observation: Lemma 2.
We have (0, ∞) = x−1 ◦ xq−2 as permutations of P1 (Fq ).
Proof. Note that xq−2 = x−1 for any nonzero element x ∈ Fq , so the map x−1 ◦ xq−2 fixes each element in F∗q . It is not hard to verify directly that x−1 ◦ xq−2 exchanges the remaining two points 0 and ∞. □ Next, we need to know the conjugations of a given 2-cycle (b, ∞) with b ∈ Fq by a degree-one polynomial x − a and by the monomial xq−2 respectively. For this purpose let us give first the following fact: Lemma 3. u, v ∈ X .
The relation f ◦ (u, v ) = (f (u), f (v )) ◦ f holds for any injective map f of sets from X to Y and any two distinct
Proof. Both f ◦ (u, v ) and (f (u), f (v )) ◦ f send x to f (x) for any element x ∈ X \ {u, v}, while both of them send u to f (v ) and send v to f (u). □ Corollary 4.
The following identities hold as permutations of P1 (Fq ):
• (x − a) ◦ (b, ∞) = (b − a, ∞) ◦ (x − a) for any a, b ∈ Fq , • xq−2 ◦ (b, ∞) = (bq−2 , ∞) ◦ xq−2 for any b ∈ Fq . Proof. The results follows from Lemma 3 by taking X = Y := P1 (Fq ), u := b, v := ∞, letting f (x) := x − a and f (x) := xq−2 respectively. □ Corollary 4 implies particularly that xq−2 ◦ (0, ∞) = (0, ∞) ◦ xq−2 , which is just equal to x−1 by Lemma 2. Moreover, it is interesting to note that the composition of any two of the three permutations x−1 , xq−2 , (0, ∞) of P1 (Fq ) is equal to the third one. In other words, the subgroup of permutations of P1 (Fq ) with q > 2 generated by x−1 , xq−2 , and (0, ∞) is exactly the Klein four-group. After conjugating (0, ∞) by the translation x + a, we can obtain the following expression of a 2-cycle of the form (a, ∞) with a ∈ Fq :
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Corollary 5.
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(a, ∞) = (x−1 + a) ◦ xq−2 ◦ (x − a) holds for any a ∈ Fq .
Proof. By applications of Corollary 4 and Lemma 2, we have (a, ∞) = (x + a) ◦ (0, ∞) ◦ (x − a) = (x + a) ◦ x−1 ◦ xq−2 ◦ (x − a), which is equal to (x−1 + a) ◦ xq−2 ◦ (x − a) obviously.
□
3. Proof of Theorem 1 For a fixed positive integer k, let us define maps F : Fkq → Fkq and G : Fkq → Fkq as follows, which will be used to illustrate the recipes in Theorems 7 and 8 respectively. Define F : Fkq → Fkq by sending (a1 , a2 , . . . , ak ) to (b1 , b2 , . . . , bk ), where bi := ci,i for 1 ≤ i ≤ k, and for any fixed q−2
1 ≤ i ≤ k we define ci,j with 0 ≤ j ≤ i inductively by ci,0 := 0 and ci,j := ci,j−1 + ai−j+1 . 1 In order to define the map G : Fkq → Fkq , we need to use the map Φℓ : Fℓq → Fℓ− for each ℓ with 2 ≤ ℓ ≤ k, which q is defined to send (e1 , e2 , . . . , eℓ ) to ((e2 − e1 )q−2 , (e3 − e1 )q−2 , . . . , (eℓ − e1 )q−2 ). For ease of notation, we will drop the subscript ℓ and write Φ for Φℓ when the value of ℓ is clear from context. Define G : Fkq → Fkq by sending (b1 , b2 , . . . , bk ) to (a1 , a2 , . . . , ak ), where for 1 ≤ i ≤ k we denote by ai the first entry of Φ i−1 (b1 , b2 , . . . , bk ), in which the map Φ i−1 : Fkq → Fqk−i+1 is really the composition Φk−i+2 ◦ Φk−i+3 ◦ · · · ◦ Φk with the convention that Φ 0 is the identity map on Fkq . More precisely, if we write
Φ i−1 (b1 , b2 , . . . , bk ) = (d1,i−1 , d2,i−1 , . . . , dk−i+1,i−1 ) for 1 ≤ i ≤ k, then by definition we have (a1 , a2 , . . . , ak ) := (d1,0 , d1,1 , . . . , d1,k−1 ). Theorem 6. on Fkq .
The maps F : Fkq → Fkq and G : Fkq → Fkq are inverses of one another. In particular, both F and G are bijections
Proof. It suffices to show that G ◦ F is the identity map on Fkq , since then F is injective and G is surjective, but since each of F and G is a map on Fkq , it follows that F and G are bijective, whence F and G are inverse bijections since G ◦ F is the identity map on Fkq again. Now, let us show that G ◦ F is the identity map on Fkq . Given a1 , a2 , . . . , ak in Fq , recall that for 1 ≤ i ≤ k the elements q−2
ci,j with 0 ≤ j ≤ i are defined inductively by ci,0 = 0 and ci,j = ci,j−1 + ai−j+1 . Put bj = cj,j for 1 ≤ j ≤ k, so that by the definition of F we get F (a1 , a2 , . . . , ak ) = (b1 , b2 , . . . , bk ). For 1 ≤ i ≤ k let us write
Φ i−1 (b1 , b2 , . . . , bk ) = (d1,i−1 , d2,i−1 , . . . , dk−i+1,i−1 ), then by the definition of G we have G(b1 , b2 , . . . , bk ) = (d1,0 , d1,1 , . . . , d1,k−1 ). We will show by induction on j that di,j = ci+j,i holds for all i, j with 0 ≤ j ≤ k − 1 and 1 ≤ i ≤ k − j. The base case j = 0 says that di,0 = ci,i for 1 ≤ i ≤ k, which is true since both sides equal bi . Inductively, if 1 ≤ j ≤ k − 1, then for each 1 ≤ i ≤ k − j we have di,j = (di+1,j−1 − d1,j−1 )q−2 = (ci+j,i+1 − cj,1 )q−2
=(ci+j,i+1 − aj )q−2 = (ciq+−j,2i )q−2 = ci+j,i , which concludes the induction. Thus G ◦ F (a1 , a2 , . . . , ak ) = G(b1 , b2 , . . . , bk )
=(d1,0 , d1,1 , . . . , d1,k−1 ) = (c1,1 , c2,1 , . . . , ck,1 ) = (a1 , a2 , . . . , ak ). So G ◦ F is the identity map on Fkq , whence as explained above it follows that F and G are inverse bijections on Fkq . Theorem 7.
For any a1 , a2 , . . . , ak in Fq , denote
(b1 , b2 , . . . , bk ) := F (a1 , a2 , . . . , ak ),
□
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then xq−2 ◦ (x − ak ) ◦ xq−2 ◦ (x − ak−1 ) ◦ · · · ◦ xq−2 ◦ (x − a1 )
=ν (x) ◦ (b1 , ∞) ◦ (b2 , ∞) ◦ · · · ◦ (bk , ∞), where
ν (x) := x−1 ◦ (x − ak ) ◦ x−1 ◦ (x − ak−1 ) ◦ · · · ◦ x−1 ◦ (x − a1 ) is a degree-one rational function in Fq (x). Proof. We prove it by induction on k. The base case k = 1 says xq−2 ◦ (x − a1 ) = x−1 ◦ (x − a1 ) ◦ (b1 , ∞) where b1 = F (a1 ) = a1 . This identity follows directly from Corollary 5. Inductively, suppose k > 1 and a1 , a2 , . . . , ak are q−2 in Fq . By definition, for 1 ≤ i ≤ k we have ci,0 = 0 and ci,j = ci,j−1 + ai−j+1 with 1 ≤ j ≤ i. Write bi = ci,i for 1 ≤ i ≤ k, so that (b1 , b2 , . . . , bk ) = F (a1 , a2 , . . . , ak ). For ease of expression, let us denote
λi := x−1 ◦ (x − ak ) ◦ xq−2 ◦ (x − ak−1 ) ◦ xq−2 ◦ · · · ◦ xq−2 ◦ (x − ai ) for 1 ≤ i ≤ k, and denote
ρi := xq−2 ◦ (x − ai ) ◦ xq−2 ◦ (x − ai−1 ) ◦ · · · ◦ xq−2 ◦ (x − a1 ) for 0 ≤ i ≤ k where by convention ρ0 = x. Write
σ := xq−2 ◦ (x − ak ) ◦ xq−2 ◦ (x − ak−1 ) ◦ · · · ◦ xq−2 ◦ (x − a1 ). Hence we have
σ = ρk =xq−2 ◦ (x − ak ) ◦ ρk−1 =x−1 ◦ (0, ∞) ◦ (x − ak ) ◦ ρk−1 =x−1 ◦ (x − ak ) ◦ (ak , ∞) ◦ ρk−1 =λk ◦ (ck,1 , ∞) ◦ ρk−1 . We claim that
λi+1 ◦ (ck,k−i , ∞) ◦ ρi = λi ◦ (ck,k−i+1 , ∞) ◦ ρi−1 for 1 ≤ i ≤ k − 1. Indeed, for each i with 1 ≤ i ≤ k − 1, we have (ck,k−i , ∞) ◦ ρi
=(ck,k−i , ∞) ◦ xq−2 ◦ (x − ai ) ◦ ρi−1 =xq−2 ◦ (ckq,−k−2 i , ∞) ◦ (x − ai ) ◦ ρi−1 =xq−2 ◦ (x − ai ) ◦ (ck,k−i+1 , ∞) ◦ ρi−1 . Thus by prepending λi+1 to these permutations we get
λi+1 ◦ (ck,k−i , ∞) ◦ ρi =λi+1 ◦ xq−2 ◦ (x − ai ) ◦ (ck,k−i+1 , ∞) ◦ ρi−1 =λi ◦ (ck,k−i+1 , ∞) ◦ ρi−1 , which concludes the proof of the claim. So we have
σ = λk ◦ (ck,1 , ∞) ◦ ρk−1 = λk−1 ◦ (ck,2 , ∞) ◦ ρk−2 = · · · = λ1 ◦ (ck,k , ∞) ◦ ρ0 , which by definition is equal to x−1 ◦ (x − ak ) ◦ xq−2 ◦ (x − ak−1 ) ◦ xq−2 ◦ · · · ◦ xq−2 ◦ (x − a1 ) ◦ (bk , ∞). By inductive hypothesis, we have xq−2 ◦ (x − ak−1 ) ◦ xq−2 ◦ (x − ak−2 ) ◦ · · · ◦ xq−2 ◦ (x − a1 )
=x−1 ◦ (x − ak−1 ) ◦ x−1 ◦ (x − ak−2 ) ◦ · · · ◦ x−1 ◦ (x − a1 )◦ ◦ (b1 , ∞) ◦ (b2 , ∞) ◦ · · · ◦ (bk−1 , ∞).
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Therefore, we get
σ =x−1 ◦ (x − ak ) ◦ x−1 ◦ (x − ak−1 ) ◦ · · · ◦ x−1 ◦ (x − a1 )◦ ◦ (b1 , ∞) ◦ (b2 , ∞) ◦ · · · ◦ (bk , ∞), which concludes the induction and hence concludes the proof. Theorem 8.
□
For any b1 , b2 , . . . , bk in Fq , denote
(a1 , a2 , . . . , ak ) := G(b1 , b2 , . . . , bk ), then (b1 , ∞) ◦ (b2 , ∞) ◦ · · · ◦ (bk , ∞)
=µ(x) ◦ xq−2 ◦ (x − ak ) ◦ xq−2 ◦ (x − ak−1 ) ◦ · · · ◦ xq−2 ◦ (x − a1 ), where
µ(x) := (x + a1 ) ◦ x−1 ◦ (x + a2 ) ◦ x−1 ◦ · · · ◦ (x + ak ) ◦ x−1 is a degree-one rational function in Fq (x). Proof. Let us prove it by induction on k. The base case k = 1 says (b1 , ∞) = (x + a1 ) ◦ x−1 ◦ xq−2 ◦ (x − a1 ) where a1 = G(b1 ) = b1 , which is true by Corollary 5. For the inductive step, assume that k > 1 and b1 , b2 , . . . , bk are in Fq . Write
Φ i−1 (b1 , b2 , . . . , bk ) = (d1,i−1 , d2,i−1 , . . . , dk−i+1,i−1 ) for 1 ≤ i ≤ k, so that (a1 , a2 , . . . , ak ) = G(b1 , b2 , . . . , bk ) = (d1,0 , d1,1 , . . . , d1,k−1 ). For ease of expression, let us define
γi := (x + b1 ) ◦ x−1 ◦ (d1,1 , ∞) ◦ (d2,1 , ∞) ◦ · · · ◦ (di−1,1 , ∞) ◦ xq−2 ◦ (x − b1 ) for 1 ≤ i ≤ k, and define
θi := (bi+1 , ∞) ◦ (bi+2 , ∞) ◦ . . . ◦ (bk , ∞) for 0 ≤ i ≤ k, where by convention γ1 = (x + b1 ) ◦ x−1 ◦ xq−2 ◦ (x − b1 ) and θk = x. Denote
τ := (b1 , ∞) ◦ (b2 , ∞) ◦ · · · ◦ (bk , ∞). Hence by Corollary 5 we have
τ = θ0 = (b1 , ∞) ◦ θ1 = (x + b1 ) ◦ x−1 ◦ xq−2 ◦ (x − b1 ) ◦ θ1 = γ1 ◦ θ1 . We claim that γi ◦ θi = γi+1 ◦ θi+1 for 1 ≤ i ≤ k − 1. Indeed, for each i with 1 ≤ i ≤ k − 1, we have xq−2 ◦ (x − b1 ) ◦ θi
=xq−2 ◦ (x − b1 ) ◦ (bi+1 , ∞) ◦ θi+1 =xq−2 ◦ (bi+1 − b1 , ∞) ◦ (x − b1 ) ◦ θi+1 =(di,1 , ∞) ◦ xq−2 ◦ (x − b1 ) ◦ θi+1 . Thus by prepending (x + b1 ) ◦ x−1 ◦ (d1,1 , ∞) ◦ (d2,1 , ∞) ◦ · · · ◦ (di−1,1 , ∞) to these permutations, we get
γi ◦ θi =(x + b1 ) ◦ x−1 ◦ (d1,1 , ∞) ◦ (d2,1 , ∞) ◦ · · · ◦ (di−1,1 , ∞) ◦ xq−2 ◦ (x − b1 ) ◦ θi =(x + b1 ) ◦ x−1 ◦ (d1,1 , ∞) ◦ (d2,1 , ∞) ◦ · · · ◦ (di−1,1 , ∞) ◦ (di,1 , ∞) ◦ xq−2 ◦ (x − b1 ) ◦ θi+1 =γi+1 ◦ θi+1 ,
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which concludes the proof of the claim. So we have
τ = γ1 ◦ θ1 = γ2 ◦ θ2 = · · · = γk ◦ θk which by definition is equal to (x + b1 ) ◦ x−1 ◦ (d1,1 , ∞) ◦ (d2,1 , ∞) ◦ · · · ◦ (dk−1,1 , ∞) ◦ xq−2 ◦ (x − b1 ). By inductive hypothesis, if we denote (aˆ1 , aˆ2 , . . . , aˆ k−1 ) := G(d1,1 , d2,1 , . . . , dk−1,1 ), then (d1,1 , ∞) ◦ (d2,1 , ∞) ◦ · · · ◦ (dk−1,1 , ∞) −1 =(x + aˆ1 ) ◦ x−1 ◦ (x + aˆ2 ) ◦ x−1 ◦ · · · ◦ (x + aˆ ◦ k−1 ) ◦ x
q−2 q−2 ◦ xq−2 ◦ (x − aˆ ◦ (x − aˆ ◦ (x − aˆ1 ). k−1 ) ◦ x k−2 ) ◦ · · · ◦ x
Note that for 1 ≤ i ≤ k − 1 we have ˆ ai = ai+1 since
Φ i−1 (d1,1 , d2,1 , . . . , dk−1,1 ) = Φ i (b1 , b2 , . . . , bk ). Therefore, we have
τ =(x + b1 ) ◦ x−1 ◦ (x + a2 ) ◦ x−1 ◦ (x + a3 ) ◦ x−1 ◦ · · · ◦ (x + ak ) ◦ x−1 ◦ xq−2 ◦ (x − ak ) ◦ xq−2 ◦ (x − ak−1 ) ◦ · · · ◦ xq−2 ◦ (x − a2 ) ◦ xq−2 ◦ (x − b1 ), which concludes the induction and the proof since b1 = a1 .
□
We remark that Theorem 7 gives a recipe for turning an algebraical presentation of a permutation on P1 (Fq ) of the form
µ(x) ◦ xq−2 ◦ (x − ak ) ◦ xq−2 ◦ (x − ak−1 ) ◦ · · · ◦ xq−2 ◦ (x − a1 ) into a combinatorial presentation of the same permutation of the form
ν (x) ◦ (b1 , ∞) ◦ (b2 , ∞) ◦ · · · ◦ (bk , ∞), and conversely Theorem 8 gives a recipe for turning a combinatorial presentation of a permutation on P1 (Fq ) of the form
ν (x) ◦ (b1 , ∞) ◦ (b2 , ∞) ◦ · · · ◦ (bk , ∞) into an algebraical presentation of the same permutation of the form
µ(x) ◦ xq−2 ◦ (x − ak ) ◦ xq−2 ◦ (x − ak−1 ) ◦ · · · ◦ xq−2 ◦ (x − a1 ), where a1 , a2 . . . , ak , b1 , b2 . . . , bk are elements in Fq and µ(x), ν (x) are degree-one rational functions in Fq (x). In other words, for any fixed permutation σ of P1 (Fq ), there are two natural maps F : Aσ ,k → Cσ ,k and G : Cσ ,k → Aσ ,k induced by the recipes illustrated in Theorems 7 and 8 respectively, where Aσ ,k is the set of all representations of σ of the form
σ = µ(x) ◦ xq−2 ◦ (x − ak ) ◦ xq−2 ◦ (x − ak−1 ) ◦ · · · ◦ xq−2 ◦ (x − a1 ) with a1 , a2 , . . . , ak ∈ Fq and µ(x) ∈ Fq (x) of degree one, and Cσ ,k is the set of all representations of σ of the form
σ = ν (x) ◦ (b1 , ∞) ◦ (b2 , ∞) ◦ · · · ◦ (bk , ∞) with b1 , b2 , . . . , bk ∈ Fq and ν (x) ∈ Fq (x) of degree one. Now we are ready to prove Theorem 1, which asserts the natural maps F : Aσ ,k → Cσ ,k and G : Cσ ,k → Aσ ,k induced by the recipes in Theorems 7 and 8 respectively are inverses to one another. Proof of Theorem 1. It suffices to show that both G ◦ F and F ◦ G are the identity maps, where F : Aσ ,k → Cσ ,k and G : Cσ ,k → Aσ ,k are the natural maps induced by Theorems 7 and 8 respectively. Given any algebraic representation of σ in the set Aσ ,k of the form
σ = µ(x) ◦ xq−2 ◦ (x − ak ) ◦ xq−2 ◦ (x − ak−1 ) ◦ · · · ◦ xq−2 ◦ (x − a1 ) with a1 , a2 , . . . , ak ∈ Fq and µ(x) ∈ Fq (x) of degree one, by definition F sends it to a combinatorial representation of σ in Cσ ,k of the form
σ = ν (x) ◦ (b1 , ∞) ◦ (b2 , ∞) ◦ · · · ◦ (bk , ∞)
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with b1 , b2 , . . . , bk ∈ Fq and ν (x) ∈ Fq (x) of degree one, in which (b1 , b2 , . . . , bk ) := F (a1 , a2 , . . . , ak ),
ν0 (x) := x−1 ◦ (x − ak ) ◦ x−1 ◦ (x − ak−1 ) ◦ · · · ◦ x−1 ◦ (x − a1 ), and ν (x) := µ(x) ◦ ν0 (x). This combinatorial representation is sent by G to another algebraic representation of σ in Aσ ,k of the form
σ = µ∗ (x) ◦ xq−2 ◦ (x − a∗k ) ◦ xq−2 ◦ (x − a∗k−1 ) ◦ · · · ◦ xq−2 ◦ (x − a∗1 ) with a∗1 , a∗2 , . . . , a∗k ∈ Fq and µ∗ (x) ∈ Fq (x) of degree one, in which (a∗1 , a∗2 , . . . , a∗k ) := G(b1 , b2 , . . . , bk ),
µ0 (x) := (x + a∗1 ) ◦ x−1 ◦ (x + a∗2 ) ◦ x−1 ◦ · · · ◦ (x + a∗k ) ◦ x−1 , and µ∗ (x) := ν (x) ◦ µ0 (x). By Theorem 6, the k-tuple (a∗1 , a∗2 , . . . , a∗k ) = G(b1 , b2 , . . . , bk ) = G ◦ F (a1 , a2 , . . . , ak ) equals (a1 , a2 , . . . , ak ), i.e., we have a∗i = ai for any 1 ≤ i ≤ k. Thus µ0 (x) and ν0 (x) are inverse to one another, which implies that
µ∗ (x) = ν (x) ◦ µ0 (x) = µ(x) ◦ ν0 (x) ◦ µ0 (x) = µ(x). Hence, G ◦ F is the identity map on the set Aσ ,k . Similarly, we can show that F ◦ G is the identity map on Cσ ,k . Therefore, the natural maps F : Aσ ,k → Cσ ,k and G : Cσ ,k → Aσ ,k induced by Theorems 7 and 8 respectively are inverse bijections. □ 4. Consequences Note that any permutation of Fq can be extended uniquely to a permutation of P1 (Fq ). Hence permutations of Fq can be regarded as permutations of P1 (Fq ) in this sense. By application of Theorem 1 to permutations of Fq , we get the following result: Corollary 9. Suppose f is a permutation of Fq , and then extends f to a permutation of P1 (Fq ) which fixes ∞. Then in any representation f = µ(x) ◦ xq−2 ◦ (x − ak ) ◦ xq−2 ◦ (x − ak−1 ) ◦ · · · ◦ xq−2 ◦ (x − a1 ) with a1 , a2 , . . . , ak ∈ Fq and µ(x) ∈ Fq (x) of degree one, the rational function µ(x) must be a degree-one polynomial in Fq [x]. Therefore, the set of all representations of f in the form of
µ(x) ◦ xq−2 ◦ (x − ak ) ◦ xq−2 ◦ (x − ak−1 ) ◦ · · · ◦ xq−2 ◦ (x − a1 ) with a1 , a2 , . . . , ak ∈ Fq and µ(x) ∈ Fq [x] of degree one is naturally bijective to the set of all representations of f in the form of
ν (x) ◦ (b1 , ∞) ◦ (b2 , ∞) ◦ · · · ◦ (bk , ∞) with b1 , b2 , . . . , bk ∈ Fq and ν (x) ∈ Fq (x) of degree one. Proof. Note that the degree-one rational function µ(x) ∈ Fq (x) in any representation of the extended permutation f on P1 (Fq ) in the form of f = µ(x) ◦ xq−2 ◦ (x − ak ) ◦ xq−2 ◦ (x − ak−1 ) ◦ · · · ◦ xq−2 ◦ (x − a1 ) fixes the point ∞, since all of f , xq−2 , and (x − ai ) fix ∞. Hence the first assertion follows from the fact that the degree-one rational functions which fix ∞ are precisely the degree-one polynomials. The second assertion is then a direct consequence of Theorem 1. □ The following is a classical result on permutations of Fq , which was posed as a question by Straus, and was first proved by Carlitz [3] and then proved again by Zieve [22]: Theorem 10. If q > 2 is a prime power, then every permutation of Fq is a composition of xq−2 and degree-one polynomials over Fq . Inspired by Theorem 10, Aksoy et al. [1] introduced the notion of the Carlitz rank of a permutation f of Fq , which means the smallest value of n ≥ 0 in any representation of f of the form f = θ0 ◦ xq−2 ◦ θ1 ◦ xq−2 ◦ . . . ◦ xq−2 ◦ θn
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in which each θi is a degree-one polynomial in Fq [x]. For a systematic study of the general theory of Carlitz ranks and their generalizations, see [11]. Carlitz ranks have been extensively studied in the literature, see for example the papers [1,6–9,14–17,19,20]. For the history of permutation polynomials over finite fields, see the papers [2–5,10,12,13,18,21]. As the first consequence, our results provide a new perspective on the theory of Carlitz ranks from the point of view of combinatorics, instead of from the point of view of algebra as in the literature previously. More precisely, by Corollary 9 the Carlitz rank of a permutation f of Fq is exactly the smallest integer n ≥ 0 for which the uniquely extended permutation of P1 (Fq ), which is also denoted by f by abuse of language, admits a combinatorial representation of the form f = ν (x) ◦ (b1 , ∞) ◦ (b2 , ∞) ◦ · · · ◦ (bn , ∞) for some b1 , b2 , . . . , bn ∈ Fq and some ν (x) ∈ Fq (x) of degree one. Based on this new characterization of Carlitz rank, the following result on the computation of Carlitz rank has be obtained in [11]: Theorem 11. Let f = µ ◦ σ be a permutation of Fq , where µ is a degree-one rational function, and σ is a permutation of P1 (Fq ) which moves s points in Fq and has t nontrivial orbits. Define n := s + t if σ (∞) = ∞ and n := s + t − 1 if σ (∞) ̸ = ∞. Then the Carlitz rank of f is at most n, and it equals n if in addition q ≥ n + s + 2. As the second consequence, our results provide new perspective to Theorem 10. Indeed, Theorem 10 follows easily from Corollary 9 in various ways, since any permutation f of Fq has many different ways to be represented as f = (b1 , ∞) ◦ (b2 , ∞) ◦ · · · ◦ (bk , ∞) for some k ≥ 1 and some b1 , b2 , . . . , bk ∈ Fq , and by Corollary 9 any such combinatorial representation of f gives rise to an algebraic representation of f as a composition of k copies of xq−2 and degree-one polynomials over Fq , which concludes our proof for Theorem 10. More precisely, given any nontrivial permutation f of Fq , we can write f as the product of disjoint cycles of length at least two as follows: f = (b11 , b12 , . . . , b1s1 )(b21 , b22 , . . . , b2s2 ) · · · (bt1 , bt2 , . . . , btst ), where t ≥ 1 and each si ≥ 2 with 1 ≤ i ≤ t. Note that (bi1 , bi2 , . . . , bisi ) = (bi1 , ∞) ◦ [(bisi , ∞) ◦ · · · ◦ (bi2 , ∞) ◦ (bi1 , ∞)], for any 1 ≤ i ≤ t. Hence we get a representation of f of the form f = (b1 , ∞) ◦ (b2 , ∞) ◦ · · · ◦ (bn , ∞), where s := Σit=1 si , k := s + t, and denote by (b1 , b2 , . . . , bk ) the tuple ((b11 , b1s1 , . . . , b11 ), (b21 , b2s2 , . . . , b21 ), . . . , (bt1 , btst , . . . , bt1 )). By Theorem 8, if we denote (a1 , a2 , . . . , ak ) := G(b1 , b2 , . . . , bk ), then we obtain a representation of f of the form f = µ(x) ◦ xq−2 ◦ (x − ak ) ◦ xq−2 ◦ (x − ak−1 ) ◦ · · · ◦ xq−2 ◦ (x − a1 ), where
µ(x) := (x + a1 ) ◦ x−1 ◦ (x + a2 ) ◦ x−1 ◦ · · · ◦ (x + ak ) ◦ x−1 ∈ Fq (x) is a rational function of degree one. Furthermore, µ(x) is a degree-one polynomial over Fq by Corollary 9. Therefore, we obtain an algebraic representation of the given permutation f of Fq as a composition of k copies of xq−2 and k + 1 degree-one polynomials over Fq as above. Moreover, if in addition q ≥ 2s + t + 2, then this representation of f is optimal in the sense that it has the least possible copies of xq−2 , since in this case the Carlitz rank of f is exactly k by Theorem 11. In particular, by application of the above procedure to 2-cycles of the form (0, a) with a ∈ F∗q , we can recover uniformly both proofs for Theorem 10 in Carlitz [3] and Zieve [22], which involve some clever tricks and have not been related to each other previously. First, let us review briefly the proofs for Theorem 10 in Carlitz [3] and Zieve [22]. The starting point for both proofs is the same, i.e., it is enough to show the result in the special case that the permutation is a 2-cycle of the form (0, a) with a ∈ F∗q , since any permutation of Fq can be written as a product of such 2-cycles. But Carlitz [3] and Zieve [22] give different expressions of the 2-cycle (0, a) as compositions of xq−2 and degree-one polynomials. More precisely, Carlitz [3] relies on the mysterious observation that (0, a) = (−a2 x) ◦ xq−2 ◦ (x − a) ◦ xq−2 ◦ (x + 1/a) ◦ xq−2 ◦ (x − a),
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but does not explain how it could be found; Zieve [22] observes that (0, a) = (−ax + a) ◦ xq−2 ◦ (−x + 1) ◦ xq−2 ◦ (−x + 1) ◦ xq−2 ◦ (x/a), which follows from the clever combination of Lemma 2 and the fact that the degree-one rational function 1 − x−1 induces an order-three permutation of P1 (Fq ) with a 3-cycle (∞, 1, 0). Now, we apply our procedure illustrated above to 2-cycles (0, a) with a ∈ F∗q to recover uniformly both of the above two observations in Carlitz [3] and Zieve [22] respectively. Indeed, for any a ∈ F∗q , there are two equally natural ways to express the 2-cycle (0, a) as a product of some 2-cycles of the form (b, ∞) with b ∈ Fq . More precisely, by our procedure the 2-cycle (0, a) can be expressed naturally as either (0, a) = (a, ∞) ◦ (0, ∞) ◦ (a, ∞) or (0, a) = (0, ∞) ◦ (a, ∞) ◦ (0, ∞). By some computation as explained above, the first identity leads exactly to the mysterious observation in Carlitz [3], and the second one gives (0, a) = (−a2 x + a) ◦ xq−2 ◦ (x + a) ◦ xq−2 ◦ (x − 1/a) ◦ xq−2 ◦ x, which is the same as the observation in Zieve [22] as polynomials. To see this it is enough to normalize all except the first degree-one polynomials to be monic in the observation discovered by Zieve [22]. Finally, we emphasize that it is of essential importance to introduce the extra point ∞ and to work over P1 (Fq ) instead of Fq , although both Carlitz’s original Theorem 10 and the notion of Carlitz rank are only about permutations of Fq and appear to have nothing to do with ∞. It is clear that neither of the above two consequences can be possibly obtained by working over Fq without introducing ∞. Declaration of competing interest The author declares that he has no conflicts of interest to this work. References [1] E. Aksoy, A. Çeşmelioğlu, W. Meidl, A. Topuzoğlu, On the Carlitz rank of permutation polynomials, Finite Fields Appl. 15 (2009) 428–440. [2] E. Betti, Sopra la risolubilità per radicali delle equazioni algebriche irriduttibili di grado primo, Ann. Sci. Mat. Fisiche 2 (1851) 5–19, (=Opere Matematiche, 1, 17–27). [3] L. Carlitz, Permutations in a finite field, Proc. Amer. Math. Soc. 4 (1953) 538. [4] L. Carlitz, A note on permutations in an arbitray field, Proc. Amer. Math. Soc. 14 (1963) 101. [5] L. Carlitz, Permutations in finite fields, Acta Sci. Math. (Szeged) 24 (1963) 196–203. [6] A. Çeşmelioğlu, A representation of permutations with full cycle, arXiv:1005.2019v1. [7] A. Çeşmelioğlu, W. Meidl, A. Topuzoğlu, Enumeration of a class of sequences generated by inversions, in: Coding and Cryptology, World Sci. Publ., Hackensack, NJ, 2008, pp. 44–57. [8] A. Çeşmelioğlu, W. Meidl, A. Topuzoğlu, On the cycle structure of permutation polynomials, Finite Fields Appl. 14 (2008) 593–614. [9] A. Çeşmelioğlu, W. Meidl, A. Topuzoğlu, Permutations of finite fields with prescribed properties, J. Comput. Appl. Math. 259 (part B) (2014) 536–545. [10] L.E. Dickson, The analytic representation of substitutions on a power of a prime number of letters, with a discussion of the linear group, Ann. Math. 11 (1896–1897) 65–120, and 161–183. [11] Z. Ding, M.E. Zieve, Carlitz ranks and approximate rational functions, 2019, in preparation. [12] K.D. Fryer, A class of permutation groups of prime degree, Canad. J. Math. 7 (1955) 24–34. [13] K.D. Fryer, Note on permutations in a finite field, Proc. Amer. Math. Soc. 6 (1955) 1–2. [14] D. Gomez-Perez, A. Ostafe, A. Topuzoğlu, On the Carlitz rank of permutations of Fq and pseudorandom sequences, J. Complexity 30 (2014) 279–289. [15] L. Işik, On complete mappings and value sets of polynomials over finite fields (Ph.D. thesis), Sabanci University, 2015. [16] L. Işik, A. Topuzoğlu, A note on value sets of polynomials over finite fields, 2017, arXiv:1701.06158v1. [17] L. Işik, A. Topuzoğlu, A. Winterhof, Complete mappings and Carlitz rank, Des. Codes Cryptogr. 85 (2017) 121–128. [18] R. Lidl, H. Niederreiter, Finite Fields, second ed., in: Encyclopedia Math. Appl., vol. 20, Cambridge Univ. Press, New York, 1997. [19] W. Meidl, A. Topuzoğlu, On the inversive pseudorandom number generator, in: Recent Developments in Applied Probability and Statistics, Physica, Heidelberg, 2010, pp. 103–125. [20] A. Topuzoğlu, The Carlitz rank of permutations of finite fields: A survey, J. Symbolic Comput. 64 (2014) 53–66. [21] C. Wells, Generators for groups of permutation polynomials over finite fields, Acta Sci. Math. Szeged. 29 (1968) 167–176. [22] M.E. Zieve, On a theorem of Carlitz, J. Group Theory 17 (2014) 667–669.