Enumeating pemutation polynomials Theodoulos Gaefalakis a,1, Giogos Kapetanakis a,, a Depatment of Mathematics and Applied Mathematics, Univesity of Cete, 70013 Heaklion, Geece Abstact We conside thoblem of enumeating polynomials ove F q, that have cetain coefficients pescibed to given values and pemute cetain substuctues of F q. In paticula, we ae inteested in the goup of -th oots of unity and in the submodules of F q. We employ the techniques of Konyagin and Pappaladi to obtain esults that ae simila to thei esults in [Finite Fields and thei Applications, 11):6 37, 006]. As a consequence, wove conditions that ensue the existence of low-degeemutation polynomials of the mentioned substuctues of F q. Keywods: finite fields, pemutation polynomials 010 MSC: 11T06, 11T3 1. Intoduction Let q = p t, whe is a pime and t is a positive intege. A polynomial ove the finite field F q is called a pemutation polynomial if it induces a pemutation on F q. The study of pemutation polynomials goes back to the wok of Hemite [6], Dickson [5], and subsequently Calitz [3] and othes. Recently, inteest in pemutation polynomials has been enewed due to applications they have found in coding theoy, cyptogaphy and combinatoics. We efe to Chapte 7 of [10] fo backgound on pemutation polynomials, as well as an extensive discussion on the histoy of the subect. In a ecent wok, Coulte, Hendeson and Matthews [4] pesent a new constuction of pemutation polynomials. Thei method equies a polynomial that pemutes the goup of -th oots of unity, µ, whee q 1, and an auxiliay function T which contacts F q to µ 0} and has some additional lineaity popety. This idea was genealized by Akbay, Ghioca and Wang []. Coesponding autho Email addesses: tgaef@uoc.g Theodoulos Gaefalakis), gnkapet@gmail.com Giogos Kapetanakis) 1 Tel.: +30 810 393845, Fax: +30 810 393881 Tel.: +30 810 39374, Fax: +30 810 393881 Pepint submitted to Finite Fields and thei Applications Decembe 9, 016
In diffeent line of wok, Konyagin and Pappaladi [7, 8] count themutation polynomials that have given coefficients equal to zeo. Given a pemutation σ SF q ), thee exists a uniquolynomial in f σ F q [X] of degee at most q such that f σ c) = σc) fo all c F q. Fo any 0 < k 1 < < k d < q 1, they define q k 1,..., k d ) to be the numbe of pemutations σ such that the coesponding polynomial f σ has the coefficients of X ki, 1 i d, equal to zeo and pove the following main esult. Theoem 1.1 [8], Theoem 1). qk 1,..., k d ) q! 1 + 1 ) q q k 1 1)q) q/. e In paticula, this implies that thee exist such pemutations, given that q!/ > 1 + e 1/ ) q q k 1 1)q) q/. Akbay, Ghioca and Wang [1] shapened this esult by enumeating pemutation polynomials of pescibed shape, that is, with a given set of non-zeo monomials. In thesent wok, we conside thoblem of enumeating polynomials ove F q, that have cetain coefficients fixed to given values, and pemute cetain substuctues of F q, namely the goup of -th oots of unity and submodules of F q and pove the following theoems. Theoem 1.. If!/ [q 1) k 1 )] / 1 + e 1/ ), then thee exists a polynomial of F q [X] of degee at most 1, that pemutes µ, the -th oots of unity, with the coefficients of X ki equal to a i F q, fo i = 1,..., d and 0 < k 1 < < k d <, whee q 1 and q is the minimum diviso of q with q 1. Theoem 1.3. Let F be a pope subfield of F q. Suppose!/ q / k 1 1) / 1 + e 1/ ), then thee exists a polynomial of F q [X] that pemutes F, an F [X]-submodule of F q, with its coefficients of X ki equal to a i F q, fo i = 1,..., d and 0 < k 1 < < k d <, whee = n = F, q = ρ and ρ is the ode and n is the degee of the Ode of F. We employ the techniques of Konyagin and Pappaladi to obtain esults that ae simila to those in [8]. In paticula, Theoems 1. and 1.3 can be viewed as the analoges of Theoem 1.1 fo oots of unity and submodules espectively, while they also imply the existence of low-degeolynomials that pemute these substuctues of F q, see Coollaies.1 and 3.1.. Enumeation of polynomials that pemute oots of unity Let q 1 and σ Sµ ) be a pemutation of µ. We define the polynomial f σ X) = 1 σc)g c X), 1) c µ
whee g c X) = 1 =0 c X, fo c µ. It is clea that g c c) = and g c x) = 0 fo all x µ \ c}, hence f σ ω) = σω), fo evey ω µ. Given d integes 0 < k 1 < < k d <, we denote q k, a) = σ Sµ ) the coefficient of X ki of f σ is a i, 1 i d }, whee k = k 1,..., k d ) and a = a 1,..., a d ) F d q. Fom 1), we see that the -th coefficient of f σ is equal to some a F q if and only if c µ c σc) = a, so we have q k, a) = σ Sµ ) } c ki σc) = a i, 1 i d. c µ Fo any S µ, define the sets } A S = f : µ S c ki fc) = a i, 1 i d, c µ } B S = f : µ S f is suective, c ki fc) = a i, 1 i d. c µ Also, define AS) = A S and BS) = B S. It is not had to see that since AM) = T M BT ), fo evey M µ, we have that Fo M = µ, the above implies BM) = 1) M T AT ). ) T M q k, a) = S µ 1) S AS). 3) Recall that q = p t. Set u) := e πiu/p and Tx) the absolute tace of x F q, i.e. Tx) := x + x p + + x pt 1 fo x F q. Futhe, let q be the smallest powe of p such that q 1, i.e. F q is the smallest subfield of F q containing µ. If 3
S µ, then AS) = 1 whee α 1,...,α d ) Fq d f:µ S = 1 α 1,...,α d ) Fq d f:µ S c µ = 1 α 1,...,α d ) F d q d T α i a i + ))) c ki fc) c µ fc) )) d α ic ki c µ T T )) d α ia i T t )) d α ic ki T d α ia i )) = S + R S, 4) R S qd 1 = qd 1 max α 1,...,α d ) F d q \0} max α 1,...,α d ) F\0} c µ c µ Moeove, the AM-GM inequality implies c µ e p T t )) d α i c ki 1 T T t )) d α ic ki )) d α ia i )) d e p T t α i c ki. c µ e p T t 1 T t q )) d α i c ki )) d α i c ki 1 k 1 ) Ttu)) u F q With the help of the well-known identity, see [9, Chapte 3], we eventually get that T t c µ Ttu)) u F q / = q S, 5) )) d ) / q 1) k1 ) S α i c ki, 4. /
which implies R S qd 1 ) / ) / q 1) k1 ) S q 1) k1 ) S <. 6) By woking similaly as in Eq. ), but by consideing the mappings µ µ, we conclude that S µ 1) S S =!, that combined with Equations 3), 4) and 6) and the fact that e / 1, since 1 + x e x fo all x, we get qk, a)! ) / q 1) < k1 ) ) / Summing up, we havoved that =0 ) / q 1) k1 ) ) e / 1 ) / = [q 1) k 1 )] / =0 =0 = [q 1) k 1 )] / 1 + e 1/ ). ) e 1/ ) q k, a) >! [q 1) k 1)] / 1 + e 1/ ), which implies the Theoem 1.. If we apply this esult in the case k b = 1, k b 1 = 1 1,..., k 1 = b and a i = 0 fo all i, then we end up with the following inteesting consequence. Coollay.1. With the same assumptions as in Theoem 1., if!/q b bq 1)1 + e 1/ ), then thee exists a polynomial of F q of degee less than b that pemutes µ. 3. Enumeation of polynomials that pemute additive submodules Thoughout this section, we see F q as a F [X]-module, whee F is a pope subfield of F q, unde the action f x = k i=0 f ix qi fo f = k i=0 f ix i F [X] and x F q. Futhemoe, it follows diectly fom the omal Basis Theoem, see [10, Theoem.35], that F q is a cyclic F [X]-module. Let F be an F [X]-submodule of F q, whee := F = n q. Since F [X] is a pincipal ideal domain and F is a F [X]-submodule of F q, which is cyclic, it 5
follows that F will be cyclic as well, see [11, Theoem 6.3]. As a consequence, thee exists some monic f F [X], of degee n, with f X m 1, such that F = x F q f x = 0}, which is known as the Ode of F. Also, fo evey x F we have that n f i x i 1 0, if x 0, = f 0, if x = 0, i=0 while f 0 0, since f X m 1. ow, fo σ SF) a pemutation of F, we define f σ X) = 1 n σc) f i X c) i 1 7) f 0 and it is clea that f σ ω) = σω) fo evey ω F. Given d integes 0 < k 1 < < k d < and a 1,..., a d ) F d q, we denote q k, a) = σ SF) the coefficient of X ki of f σ is a i, 1 i d }. Fom 7), we deduce that the -th coefficient of f σ is a if and only if hence i=0 q k, a) = σ SF) i=0 n i ) 1 f i c) i 1 σc) = f 0 a, n i ) 1 f i c i 1 k σc) = f 0 a, 1 d}. i=0 Fo any S F, define the sets A S = g : F S k } n F i c i 1 k gc) = f 0 a, 1 d, i=0 B S = g A S g is suective}, whee F i stands fo ) i 1 k fi. Define AS) = A S and BS) = B S. As with Eq. ), we can show that AM) = T M BT ), fo evey M F, hence q k, a) = S F 1) S AS). 8) Futhemoe, let ρ be the ode of f, i.e. ρ is minimal such that f X ρ 1 and let q := ρ. It follows that F q is the smallest subfield of F q containing F. Fo 6
S F, as in the case of Equation 4), we have AS) = 1 T gc) d =1 α )) n i=0 F ic i 1 k d )) α 1,...,α d ) Fq d g:f S T =1 f 0α a = 1 T t d )) n =1 i=0 α F i c i 1 k d )) T =1 f 0α a whee α 1,...,α d ) F d q = S + R S, 9) R S qd 1 = qd 1 max α 1,...,α d ) F d q \0} max α 1,...,α d ) F\0} T T T t t d )) n =1 i=0 α F i c i 1 k d )) =1 f 0α a d =1 i=0 n α F i c i 1 k. Also, the AM-GM inequality yields d n T t α F i c i 1 k =1 i=0 1 d n / T t α F i c i 1 k 1 q T t =1 i=0 d =1 i=0 1 1 k 1 ) e p Ttu)) u F q With the help of 5), we show that d n T t α F i c i 1 k that, in tun, yields R S qd 1 =1 i=0 n / α F i c i 1 k /. q 1 k ) / S ), q 1 k ) / S ) < q 1 k ) / S ). 10) 7
ow, as in the case of the oots of unity, it is clea that 1) S S =!, S F which combined with Equations 8), 9) and 10) and the fact that e / 1, gives qk, a)! < q/ 1 k ) / =0 q / 1 k ) / =0 ) / = q / k 1 1) / 1 + e 1/ ). To sum up, in this section woved that ) e / 1) / q k, a) >! q/ k 1 1) / 1 + e 1/ ) which implies Theoem 1.3. By applying this fo k b = 1, k b 1 = 1 1,..., k 1 = b and a i = 0 fo all i, we end up with the following. Coollay 3.1. With the same assumptions as in Theoem 1.3, if! qb 1)1 + e 1/ ), then thee exists a polynomial of F q of degee less than b that pemutes F. Aknowledgments The authos would like the eviewe fo his/he suggestions and coections. Refeences [1] A. Akbay, D. Ghioca and Q. Wang. On pemutation polynomials with pescibed shape. Finite Fields Appl., 15:195 06, 009. [] A. Akbay, D. Ghioca and Q. Wang. On constucting pemutations of finite fields. Finite Fields Appl., 17:51 67, 011. [3] L. Calitz. Pemutations in a finite field. Poc. Ame. Math. Soc., 4:538, 1953. [4] R. Coulte, M. Hendeson and R. Matthews. A note on constucting pemutation polynomials. Finite Fields Appl., 15:553 557, 009. 8
[5] L. Dickson. The analytic epesentation of substitutions on a powe of a pime numbe of lettes with a discussion of the linea goup. Ann. of Math., 11:65 10, 161 183, 1897. [6] C. Hemite. Su les fonctions de sept lettes. C.R. Acad. Sci. Pais, 57:750 757, 1863. [7] S. Konyagin and F. Pappaladi. Enumeating pemutation polynomials ove finite fields by degee. Finite Fields Appl., 84):548 553, 00. [8] S. Konyagin and F. Pappaladi. Enumeating pemutation polynomials ove finite fields by degee II. Finite Fields Appl., 11):6 37, 006. [9] S. Konyagin and I. Shpalinski. Chaacte Sums with Exponential Functions and thei Applications. Cambidge Univesity Pess, Cambidge, 004. [10] R. Lidl and H. iedeeite. Finite Fields. Addison-Wesley, Reading, Mass., 1983. [11] S. Roman. Advanced Linea Algeba. Spinge-Velag, ew Yok, 008. 9