Enumeration of a special class of irreducible polynomials in characteristic 2
EENUMERATION OF A SPECIAL CLASS OF IRREDUCIBLEPOLYNOMIALS IN CHARACTERISTIC 2
ALP BASSA ¶ AND RICARDO MENARES † Abstract. A -polynomials were introduced by Meyn and play an important role in the iterativeconstruction of high degree self-reciprocal irreducible polynomials over the field F , since theyconstitute the starting point of the iteration. The exact number of A -polynomials of eachdegree was given by Niederreiter. Kyuregyan extended the construction of Meyn to arbitraryeven finite fields. We relate the A -polynomials in this more general setting to inert places ina certain extension of elliptic function fields and obtain an explicit counting formula for theirnumber. In particular, we are able to show that, except for an isolated exception, there existA-polynomials of every degree. The Q -transform plays a prominent role in the construction of (self-reciprocal) irreduciblepolynomials. Given a polynomial f ∈ F q [ T ], its Q -transform is given by f Q ( T ) := T deg f · f (cid:16) T + 1 T (cid:17) . Then, f Q is a self-reciprocal polynomial of degree 2 · deg f . Clearly for f Q to be irreducible anecessary condition is that f is irreducible. In characteristic 2, a simple sufficient and necessarycondition for the irreducibility of f Q in terms of the coefficients of f was established by Meyn([Mey90], Theorem 6). More surprising is the fact that it is even possible to devise criteria toensure that irreducibility is preserved under repeated applications of the Q -transform, hencegiving an infinite sequence of self-reciprocal irreducible polynomials of increasing degree. Start-ing with an irreducible polynomial f ∈ F q [ T ], we iteratively define a sequence of polynomials f m ∈ F q [ T ] by f m +1 = f Qm , m ≥ . In what follows, we set q = 2 r and describe the conditions mentioned above in characteristic2. Let f ( T ) = T n + a n − T n − + . . . + a T + a ∈ F q [ T ] be a monic irreducible polynomial ofdegree n . We say that f is an A - polynomial if T r F q / F ( a n − ) = 1 and T r F q / F ( a /a ) = 1. Then,whenever f is an A -polynomial, the polynomial f m ( T ) is irreducible of degree n m for all m .This fact was proved by Meyn [Mey90] and Varshamov [Var84], when r = 1, and generalizedlatter by Kyuregyan [Kyu02] for general r ≥ q is odd, the Q -transform behaves in a more subtle way and the results are less com-plete. In that setting, S. Cohen introduced the related R -transform, which leads to a comparableiterative construction of irreducible reciprocal polynomials [Coh92]. For a comparison of bothtransforms in terms of Galois theory see [BHMP].In this note, we provide a closed formula for the number of A -polynomials in F r [ T ] of givendegree. On the other hand, it is known that there are no A -polynomials of degree 3 in F [ T ]. Inall other cases, our formula allows us to deduce the existence of A -polynomials of each degreeover every finite field of characteristic 2. ¶ Bo˘gazi¸ci University. † Pontificia Universidad Cat´olica de Chile.Both authors were partially supported by Conicyt-MEC 80130064 grant. Alp Bassa was partially supportedby the BAGEP Award of the Science Academy with funding supplied by Mehve¸s Demiren in memory of SelimDemiren and by Bo˘gazi¸ci University Research Fund Grant Number 15B06SUP2. Ricardo Menares was partiallysupported by FONDECYT 1171329 grant. a r X i v : . [ m a t h . N T ] M a y heorem 1. We denote by A r ( n ) the number of A -polynomials in F r [ T ] of degree n . a) Write n = 2 k · m , ( m odd) and let α = − √− . Then, A r ( n ) = 14 n (cid:88) d | m µ (cid:16) md (cid:17)(cid:0) q k d + 1 − α r k d − α r k d (cid:1) . Here, µ is the M¨obius function. In particular, we have that (1) (cid:12)(cid:12)(cid:12)(cid:12) A r ( n ) − q n n (cid:12)(cid:12)(cid:12)(cid:12) ≤ σ ( m )4 n ( q n/ + 1 + 2 · rn/ ) . Here, σ ( m ) is the number of positive divisors of m . Hence, A r ( n ) ∼ q n n as n → ∞ . b) Assume ( r, n ) (cid:54) = (1 , . Then, A r ( n ) ≥ . In order to prove Theorem 1, we exploit the correspondence between irreducible polynomialsof each degree n and degree n places that are inert in a particular unramified extension of ellipticfunction fields. Then, we show that an exact count can be obtained using the corresponding L -polynomials. A very rough estimate then ensures the existence of A -polynomials of anydegree n ≥ r = 1 (cf.Remark 4). His method requires an explicit evaluation of certain Kloosterman sums attachedto additive characters, which is available only when the base field is small. By turning aroundNiederreiter’s reasoning, we obtain an archimedean evaluation of a certain weighted average ofKloosterman sums (cf. Proposition 5 below).For results and notation about algebraic function fields we refer the reader to [Sti09]. In[MR08] a related approach is applied to the problem of counting polynomials with prescribedcoefficients.1. Interpreting A -polynomials in terms of an extension of elliptic functionfields Let q = 2 r , and let F q be the finite field of q elements. Consider the rational function field F = F q ( x ) and the extensions E = F ( y ) and E = F ( y ), with y + y = x and y + y = 1 /x .Both E and E are again rational function fields. Lemma 2.
Let f ( x ) = x n + a n − x n − + . . . + a x + a ∈ F q [ x ] be a monic irreducible polynomialof degree n , with f ( x ) (cid:54) = x . We denote by P f the place of F of degree n associated to f . Then, i) P f is inert in E /F if and only if T r F q / F ( a n − ) = 1ii) P f is inert in E /F if and only if T r F q / F ( a /a ) = 1 . In particular, f is an A -polynomial if and only if P f is inert in both extensions E /F and E /F .Proof. First we prove i). If c is a root of f in F q , then c (cid:54) = 0 and a n − = T r F qn / F q ( c ). Hence bythe transitivity of the trace, the condition T r F q / F ( a n − ) = 1 is equivalent to T r F qn / F ( c ) (cid:54) = 0.By Hilbert’s Theorem 90, this happens exactly if c is not of the form γ − γ for any γ ∈ F q n . Inturn, this happens if and only if f ( y + y ) ∈ F [ y ] is irreducible. The last condition is equivalentto P f being inert in E /F , thus proving i).Part ii) follows form Part i) applied to f ∗ ( x ) = x n f (1 /x ) (cid:5) Consider the compositum E (cid:48) = E · E over F . The extension E (cid:48) /F is Galois, with Galoisgroup Z / Z × Z / Z , with E and E corresponding to the subgroups Z / Z ×{ } and { }× Z / Z ,respectively. Let E be the subfield corresponding to third subgroup H , the diagonal subgroup.Clearly E = F ( y ) with y = y + y satisfying y + y = x + 1 /x . In other words, E is the functionfield of the elliptic curve over F q with j = 1. (cid:48) = E · E H E = F ( y ) E = F ( y + y ) E = F ( y ) F = F q ( x ) Proposition 3.
Let P (cid:48) be a place of E (cid:48) above P f . We denote by G ( P (cid:48) | P f ) be the associateddecomposition group in E (cid:48) /F . Let C r ( n ) be the number if inert places of degree n in the extension E (cid:48) /E . Then, i) f is an A -polynomial if and only if G ( P (cid:48) | P f ) = H ii) We have that C r ( n ) = 2 A r ( n ) .Proof. The polynomial T corresponds to the zero P T of x . Since only the pole P ∞ of x ramifiesin the extension E /F , the places P T and P ∞ are the only places of F ramified in the extension E (cid:48) /F . Both places are ramified in E/F and in each case the place of E lying above them splitsin the extension E (cid:48) /E . The extension E (cid:48) /E is unramified. Using the Riemann-Hurwitz genusformula, we see that the genera of E and E (cid:48) are both 1. For a place P f (cid:54) = P T , P ∞ , let Z ( P f ) bethe associated decomposition group in the extension E (cid:48) /F . The place P f is inert in E /F and E /F , if and only if Z ( P f ) = H . Hence to each degree n place P f of F that is inert in E /F and E /F there correspond two places of E of degree n that are inert in E (cid:48) /E . In particular, toevery A -polynomial f there correspond two places of E of degree deg f that are inert in E (cid:48) /E . (cid:5) Enumerating A -polynomials over arbitrary even finite fields In this section we provide a proof of Theorem 1. Following Proposition 3, we need to count thenumber of inert places of E and E (cid:48) . We will achieve this task by means of their L -polynomials.All function fields can be defined already over F . Hence we consider the extension F ( x, y ) / F ( x )with y + y = x + 1 /x . Among the rational places of F ( x ), the pole and zero of x are ramified,the zero of x − F ( x, y ) / F ( x ), giving a total of 4 rational places of F ( x, y ). Theelliptic function field hence has trace − L -polynomial2 t + t + 1 = (1 − αt )(1 − αt ) with α = − √− . The L -polynomial of the constant field extension E = F q ( x, y ) with q = 2 r is hence L E ( t ) = (1 − α r t )(1 − α r t ) . The L -polynomial L E (cid:48) of E (cid:48) has to be divisible by L E and be also of degree 2, since g ( E (cid:48) ) = 1.Hence L E (cid:48) = L E . The number of degree n places for each of the function fields is given by (see[Sti09, Propositions 5.1.16 and 5.2.9])(2) B ( n ) = 1 n (cid:88) d | n µ (cid:16) nd (cid:17)(cid:0) q d + 1 − α rd − α rd (cid:1) . For even n the B ( n ) places of E (cid:48) of degree n come from the C r ( n/
2) inert places of degree n/ E and the B ( n ) − C r ( n ) splitting places of degree n (we get 2 degree n places for eachsplitting place). For odd n the B ( n ) places of E (cid:48) of degree n come only from the B ( n ) − C r ( n )splitting places of E of degree n . Hence we obtain B ( n ) = C r ( n/
2) + 2 · ( B ( n ) − C r ( n )) for n even B ( n ) = 2 · ( B ( n ) − C r ( n )) for n odd . riting n = 2 k · m for an odd integer m , we obtain(3) C r (2 k · m ) = k +1 (cid:88) i =1 i B (2 k +1 − i · m ) . Using Proposition 3, ii) and equations (2) and (3), we obtain the formula stated in the firstpart of Theorem 1. The estimate (1) is obtained by using that | α | = √ d | m satisfies d ≤ m/ E has genus 1. Hence using estimates for the number of higher degreeplaces (see for instance[Sti09, Corollary 5.2.10]) B ( n ) ≥ n with q ( n − / ( q / − ≥ r ≥ B ( n ) ≥ n . For r = 1, we need n ≥ r = 2 or3 we need n ≥ B ( n ) ≥ L -polynomial shows that in all cases except ( r, n ) = (1 , , we have B ( n ) ≥
1. Since E (cid:48) hasgenus 1, using Proposition 3, ii) and equation (3), we obtain the second assertion in Theorem1. Remark 4.
Note that for r = 1 we recover the main Theorem in [Nie90] . Indeed, use theelementary identity α t + α t = 12 t − t/ (cid:88) j =0 (cid:18) t j (cid:19) ( − t + j j , valid for all integers t ≥ . Averages of Kloosterman sums
Let q = 2 r and let χ : F q → C ∗ be an additive character. For any integer n ≥
1, let χ ( n ) : F q n → C ∗ be the additive character defined by χ ( n ) ( u ) = χ ◦ T r F qn / F ( u ) . Let K ( χ ( n ) ; a, b ) = (cid:88) α ∈ F ∗ qn χ ( n ) ( aα + bα − )be the Kloosterman sum attached to χ ( n ) . Proposition 5.
Assume χ is a non trivial additive character. Then, (4) 1 q (cid:88) u ∈ F q χ ( u ) (cid:88) a,b ∈ F qa + b = u K ( χ ( n ) ; a, b ) = (cid:88) d | n µ (cid:16) nd (cid:17) · d · A r ( d ) , for all n ≥ .Proof. let R ( n ) = { α ∈ F ∗ q n : T r F qn / F ( α ) = T r F qn / F ( α − ) = 1 } .R ∗ ( n ) = { α ∈ R ( n ) : [ F q ( α ) : F q ] = n } . We have that(5) | R ∗ ( n ) | = n · A r ( n ) , | R ( n ) | = (cid:88) d | n | R ∗ ( d ) | . Since χ is non trivial, for all u ∈ F q we have that [LN97, Corollary 5.31],1 q (cid:88) a ∈ F q χ ( ua ) = (cid:26) u = 00 otherwise.Set T ( n ) := T r F qn / F . We have that R ( n ) | = (cid:88) α ∈ F ∗ qn { T ( n ) ( α )=1 } · { T ( n ) ( α − )=1 } = (cid:88) α ∈ F ∗ qn q (cid:88) a ∈ F q χ (cid:16) ( T ( n ) ( α ) + 1) a (cid:17) (cid:88) b ∈ F q χ (cid:16) ( T ( n ) ( α − ) + 1) b (cid:17) = 1 q (cid:88) α ∈ F ∗ qn (cid:88) a,b ∈ F q χ ( a + b ) χ (cid:16) T ( n ) ( aα + bα − ) (cid:17) = 1 q (cid:88) u ∈ F q χ ( u ) (cid:88) a,b ∈ F qa + b = u (cid:88) α ∈ F ∗ qn χ ( n ) ( aα + bα − )= 1 q (cid:88) u ∈ F q χ ( u ) (cid:88) a,b ∈ F qa + b = u K ( χ ( n ) ; a, b ) . We conclude by combining this equality with the relations (5). (cid:5)
Remark 6.
When r = 1 , Niederreiter finds in [Nie90] an explicit evaluation in archimedeanterms of each individual Kloosterman sum in the LHS of (4) . It seems difficult to obtain asimilar individual evaluation for general r . Our method proceeds by interpreting the whole LHSin terms of the number of places of appropriate degree in a particular elliptic extension and thenuse the explicit knowledge of the corresponding L -polynomials to determine such quantities. References [BHMP] Alp Bassa, Emmanuel Hallouin, Ricardo Menares, and Marc Perret. Galois theory and iterative con-struction of irreducible polynomials.
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Alp Bassa. Bo˘gazic¸i University, Faculty of Arts and Sciences, Department of Mathematics,34342 Bebek, ˙Istanbul, Turkey,
E-mail address : [email protected] Ricardo Menares. Pontificia Universidad Cat´olica de Chile, Facultad de Matem´aticas, Vicu˜naMackenna 4860, Santiago, Chile.
E-mail address : [email protected]@mat.uc.cl