Irreducible polyomials with cosecutive zero coefficiets Theodoulos Garefalakis Departmet of Mathematics, Uiversity of Crete, 71409 Heraklio, Greece Abstract Let q be a prime power. We cosider the problem of the existece of moic irreducible polyomials over F q with cosecutive coefficiets fixed to zero. We show that asymptotically, there exist moic irreducible polyomials of degree over F q with roughly /3 cosecutive coefficiets fixed to zero. Key words: Irreducible polyomials, fiite fields 1 Itroductio Let F q be a fiite field with q elemets, of characteristic p. We deote by A = F q [T] the rig of polyomials over F q. It is well-kow that asymptotically, the umber of irreducible polyomials i A of degree is approximately q /. However, much less is kow about the umber, or eve the existece of irreducibles of certai form, for istace with some coefficiets fixed to give values. Give a iteger > 1, it has bee proved idepedetly by S.D. Cohe [2] ad R. Ree [13], that for all large eough q, there always is a irreducible polyomial over F q of the form T +T +a. However, much less is kow whe q is fixed ad large. I [10], T. Hase ad G.L. Mulle cojecture that give itegers > m 0 there exists a moic irreducible polyomial over F q of degree with the coefficiet of T m fixed to ay give elemet a F q. Of course, a 0 if m = 0. By cosiderig primitive polyomials with give trace, S.D. Cohe [4] proves that the cojecture is true for m = 1. I [15], D. Wa settles Email address: theo@math.uoc.gr (Theodoulos Garefalakis). Preprit submitted to Fiite Fields ad Their Applicatios 7 March 2007
the cojecture subject to the coditio that either q > 19 or 36, leavig a fiite umber of cases to be checked. By machie assisted computatios, K.H. Ham ad G.L. Mulle have verified the cojecture for these remaiig cases i [9]. Prior to this work, E.N. Kuz mi [12] determied the umber of moic irreducibles of degree 4 with the coefficiets of T 3, T 2, T fixed to give values. Special attetio is give to the case where the coefficiets of T 3 ad T 2 are zero. Similar results o the umber of polyomials of give degree of give factorizatio patter which satisfy a additioal property, such as that certai coefficiets are prescribed, have bee obtaied by S.D. Cohe i [3]. The aalogous problem for primitive polyomials has also attracted cosiderable attetio. The aalogue of Wa s result has bee proved to be true by S.D. Cohe [5]. A survey o the subject by S. Gao, J. Howell ad D. Paario, icludig experimetal results as well as some applicatios ca be foud i [8]. It is atural to ask, ad i fact to expect, that much more tha the above is true. Namely, oe would expect that irreducible polyomials exist with may coefficiets fixed to give values. The objective of this work is to show that moic irreducible polyomials of degree over F q exist with up to c cosecutive coefficiets fixed to zero, where 0 < c < 1 ad the coditio (1 3c) 2 + 8 log q is satisfied. Such irreducibles are called sparse ad have may practical applicatios, see [6,7]. The proof of the mai theorem is based o a estimate of a weighted sum, which is very similar to the oe that D. Wa cosiders. The mai tool is Weil s boud for character sums. We record the followig elemetary lemma, which will be useful later. Let f(t) = i=0 a i T i F q [T] ad deote by f (T) = i=0 a i T i its reciprocal. Lemma 1 Let f(t) = i=0 a i T i F q [T] with a a 0 0. The polyomial f is irreducible over F q if ad oly if f is irreducible over F q. 2 Character sums It is well kow that Dirichlet s theorem for primes i arithmetic progressio has a aalogue i A. Let f, h A relatively prime. We reserve the letter P to deote a moic irreducible polyomial i A. Let S (h, f) = {P A P h (mod f), deg(p) = }. We deote by π (h, f) the cardiality of S (h, f). The followig asymptotic versio of Dirichlet s theorem is well-kow, see for istace [14]. 2
Theorem 1 π (h, f) = 1 Φ(f) q ( ) q /2 + O. Here Φ(f) is the order of the group (A/fA), ad the degree of f is assumed to be costat (ot depedig o ). Let f = T m A ad h A with deg(h) m 1. The S (h, f) cotais all irreducibles with the lower m coefficiets fixed to those of h. For m fixed, Theorem 1 gives a estimate of the umber of irreducibles with the lower m coefficiets fixed to ay values. Lemma 1 the implies that the same estimate holds if the upper m coefficiets are fixed. It is well-kow that the truth of the Riema hypothesis for fuctio fields leads to effective versios of Theorem 1, see for istace [15]. The basic tools are bouds for character sums, sometimes referred to as Weil character sums. We recall here the mai otios ad results for future referece. Let χ be a character of the group (A/fA), that is, a homomorphism from (A/fA) to C. The character χ exteded to A by zero is called a Dirichlet character mod f. The trivial character, that maps all polyomials prime to f to 1 is deoted by χ o. Defie the sum c (χ) = d deg(p)=d dχ(p /d ), where the ier sum is over all moic irreducible polyomials of degree d. It will be coveiet to express, as i [15], c (χ) i terms of the vo Magolt fuctio Λ, which is defied o A as follows: Λ(h) = deg(p) if h = P e for some irreducible P ad a iteger e, ad is zero otherwise. It s rather easy to see that c (χ) = deg(h)= Λ(h)χ(h), where the sum is over all moic polyomials of degree. Further, we defie c (χ) = χ(p), deg(p)= where the sum is over moic irreducibles of degree. We deote by π the umber of irreducible polyomials of degree i A. It is well-kow that d dπ d = q. The Moebius iversio formula the implies that π = µ(d)q /d = q + d d,d>1 µ(d)q /d. 3
Sice µ(d)q /d q d 2q /2, d,d>1 0 d /2 it follows that π q 2 q/2. The followig theorem follows from the Riema hypothesis for fuctio fields, see [15]. Propositio 1 If χ χ o the (1) c (χ) (deg(f) 1)q /2, (2) c (χ) 1 (deg(f) + 1)q/2. Also, c (χ o ) = q ad c (χ o) = π. Usig Propositio 1 it is ot hard to show the followig effective versio of Dirichlet s theorem for A. Theorem 2 (Theorem 5.1 of [15]) Let f, h A, with (f, h) = 1. The π (h, f) q Φ(f) m + 1 q/2. Theorem 2, applied with f = T m, immediately implies that there always exists a moic irreducible polyomial of degree with the coefficiets of roughly /2 log q lower coefficiets fixed (provided that the costat term is ot zero). Lemma 1 the esures the existece of a moic irreducible of degree with roughly /2 log q higher coefficiets fixed. More precisely we have the followig corollary. Corollary 1 Let > m 1 be itegers satisfyig q /2 (m+1)q m. For ay β 0, β 1,...,β m 1 F q with β 0 0, there exists a moic irreducible polyomial P = T + 1 i=0 a i T i i A of degree, with a i = β i, 0 i m 1. Also, there exists a moic irreducible polyomial with a i = β i, 1 i m 1. Irreducibles of degree with roughly /2 leadig or trailig coefficiets prescribed ca by foud effectively: heuristic argumets ad experimetal results suggest that oe may prescribe up to 2 log q leadig or trailig coefficiets, ad a irreducible still exists. This set polyomials is of reasoable size polyomial i, log q ad ca therefore be searched exhaustively. This method 4
of course depeds o uprove assumptios. To the author s kowledge, there is o method that provably costructs the irreducibles of Corollary 1. We ote that Theorem 2 ad Corollary 1 are classical. Geeralizatios have bee obtaied by M. Car [1] ad C-N. Hsu [11]. It follows from these geeralizatios that irreducible polyomials of degree exist with the leadig 1 ad trailig 2 coefficiets are fixed to give values, subject to the coditio 1 + 2 is less tha roughly /2. Ufortuately, the above results do ot give us ay iformatio about irreducibles with some of the middle coefficiets fixed. 3 Irreducible polyomials with cosecutive zero coefficiets Let m > l > 1 be itegers, ad deote by H l 1 the set of moic primary polyomials of A of degree l 1. We recall that a polyomial i A is called primary if it is a power of a irreducible polyomial of A. We defie w(, m, l) = h H l 1 Λ(h) P h (mod T m ) 1, (1) where the ier sum is over moic irreducibles of degree with stated property. Provig that w(, m, l) > 0 implies that there is a moic irreducible polyomial P = T + 1 i=0 a it i of degree with the coefficiets a i = 0 for l i m 1. Theorem 3 With the above otatio, w(, m, l) ql m π q 1 < m2 1 q (+l 1)/2. PROOF. First we rewrite the sum defiig w(, m, l) as w(, m, l) = 1 Φ(T m ) χ h H l 1 Λ(h) deg(p)= χ(p) χ(h), where the first sum is over the Dirichlet characters mod T m ad the third sum is over moic irreducibles of degree. Separatig the term correspodig to χ o ad rearragig we have w(, m, l) ql 1 π Φ(T m ) = 1 Φ(T m ) χ χ o deg(p)= χ(p) Λ(h) χ(h). h H l 1 5
Therefore, w(, m, l) ql 1 π Φ(T m ) 1 Φ(T m χ(p) Λ(h) χ(h) ) χ χ o deg(p)= h H l 1 = 1 c Φ(T m ) (χ) c l 1 ( χ) χ χ o < m + 1 q/2 (m 1)q (l 1)/2 = m2 1 q (+l 1)/2, where we used the estimates of Propositio 1, ad the fact that there are Φ(T m ) distict characters of (A/T m A). Corollary 2 Let > m > l > 1 such that q +l 2m qm 4. The there exists a moic irreducible polyomial P = T + 1 i=0 a i T i A of degree such that a m 1 = = a l = 0. PROOF. From Theorem 3 it follows that w(, m, l) > ql m π (q 1) m2 1 q (+l 1)/2. Sice π q 2q/2, we have w(, m, l) > q+l m (q 1) 2q/2+l m (q 1) (m2 1)q (+l 1)/2. (2) It suffices to prove that uder the coditio of the corollary, the right-hadside is o-egative. Ideed, the right-had-side is at least ( ( )) 1 q +l m 1 m 2 q (+l 1)/2 + q (+l 1)/2 2q/2+l m q 1 ( q (+l)/2 m qm 2) q(+l 2)/2 0 where the first iequality holds sice q (+l 1)/2 2q/2+l m q 1 for 1 < l < m. Corollary 2 ca be used to show that there exist irreducible polyomials with a large umber of cosecutive coefficiets fixed to zero. 6
Corollary 3 Let 0 < c < 1 ad a atural umber such that (1 3c) 2+8 log q. The, there exists a moic irreducible polyomial of degree over F q with ay c cosecutive coefficiets, other tha the first ad the last, fixed to zero. PROOF. We apply Corollary 2 to fix m l = c (3) coefficiets. First, we observe that it suffices to cosider the case m + l. (4) The case m + l > follows from this by a applicatio of Lemma 1. We also record that the coditios i Eq. (3) ad Eq. (4) imply that 2m + c. (5) The corollary will follow if q c m qm 4 c 1 4 log q m m. Give Eq. (5), the last coditio is satisfied if 2 2 c 2 8 log q + c 3 c 2 + 8 log q. The last iequality is clearly satisfied, uder the assumptio of the corollary. The corollary shows that for ay ǫ > 0 there exist moic irreducible polyomials of degree with up to (1/3 ǫ) coefficiets fixed to zero, provided that is large eough. As a example, we show the followig corollary. Corollary 4 Let q be a prime power, ad a positive iteger. The there exists a moic irreducible polyomial of degree over F q with ay /4 coefficiets fixed to zero for prime powers 2 q 59 ad the rages for show i the table below, ad for q 60 ad 37. 7
q q q 2 266 13 59 32 43 3 155 16 55 37 41 4 119 17 53 41 40 5 100 19 51 43 40 7 81 23 48 47 39 8 75 25 47 49 38 9 70 27 45 53 38 11 64 31 44 59 37 PROOF. This is a cosequece of Corollary 3 for c = 1/4. We have to show that /4 2 + 8 log q (6) for the parameters i the table. The fuctio y q (t) = t 8 32 log q t is icreasig for t 32/ logq. For various values of q, we compute the sigle zero of y q (t) umerically, ad coclude that for t greater of equal to the zero y q (t) 0. The rages of t for each value of q are show i the table above. Sice y q (t) y q (t) for q q, we coclude that Eq. (6) holds for ay q 60 ad 38. 4 Cocludig Remarks We have proved that there exist moic irreducible polyomials of degree over F q with roughly /3 coefficiets fixed to zero. This is oly a partial extesio of the result of D. Wa [15], which shows that (uder the mild techical coditio that either q > 19 or 36) there exist moic irreducible of degree with ay oe coefficiet ca be fixed to ay value. It would be iterestig to exted the preset result, ad show that for large eough, there exist moic irreducibles of degree with roughly /3 coefficiets fixed to ay values. Ackowledgemets I would like to thak the aoymous reviewers for providig helpful commets ad for poitig out related work. 8
Refereces [1] M. Car. Distributio des polyomes irreductibles das F[t]. Acta Arith., 88:141 153, 1999. [2] S.D. Cohe. The distributio of polyomials over fiite fields. Acta Arith., 17:255 271, 1970. [3] S.D. Cohe. Uiform distributio of polyomials over fiite fields. J. Lodo Math. Soc., 2(6):93 102, 1972. [4] S.D. Cohe. Primitive elemets ad polyomials with arbitrary trace. J. America Math. Soc., 83:1 7, 1990. [5] S.D. Cohe. Primitive polyomials with a prescribed coefficiet. Fiite Fields ad Applicatios, 12(3):425 491, 2006. [6] D. Coppersmith. Fast evaluatio of logarithms i fields of characteristic two. IEEE Tras. Iform. Theory, IT-30:587 594, 1984. [7] S. Gao. Elemets of provable high orders i fiite fields. Proc. America Math. Soc., 127:1615 1623, 1999. [8] S. Gao, J. Howell, ad D. Paario. Irreducible polyomials of give forms. Cotemporary Mathematics, 225:43 53, 1999. [9] K.H. Ham ad G.L. Mulle. Distributio of irreducible polyomials of small degrees over fiite fields. Math. Comp., 67(221):337 341, 1998. [10] T. Hase ad G.L. Mulle. Primitive polyomials over fiite fields. Math. Comp., 59:639 643, 1992. [11] C-N. Hsu. The distributio of irreducible polyomials i F q [t]. J. Number Theory, 61(1):85 96, 1996. [12] E.N. Kuz mi. Irreducible polyomials over fiite fields i. Algebra ad Logic, 33(4):216 232, 1994. [13] R. Ree. Proof of a cojecture of S. Chowla. J. Number Theory, 3(2):210 212, 1971. [14] M. Rose. Number theory i fuctio fields. Spriger Verlag, 2002. [15] D. Wa. Geerators ad irreducible polyomials over fiite fields. Math. Comp., 66(219):1195 1212, 1997. 9