THE NUMBER OF IRREDUCIBLE POLYNOMIALS AND LYNDON WORDS WITH GIVEN TRACE

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1 SIM J. DISCRETE MTH. Vol. 14, No. 2, pp c 2001 Society for Iustrial a pplie Mathematics THE NUMBER OF IRREDUCIBLE POLYNOMILS ND LYNDON WORDS WITH GIVEN TRCE F. RUSKEY, C. R. MIERS, ND J. SWD bstract. The trace of a egree polyomial f(x over GF (q is the coefficiet of x 1. Carlitz [Proc. mer. Math. Soc., 3 (1952, pp ] obtaie a expressio I q(, t for the umber of moic irreucible polyomials over GF (q of egree a trace t. Usig a ifferet approach, we erive a simple explicit expressio for I q(, t. If t > 0, I q(, t = ( µ(q / /(q, where the sum is over all ivisors of which are relatively prime to q. This same approach is use to cout L q(, t, the umber of q-ary Lyo wors whose characters sum to t mo q. This umber is give by L q(, t = ( gc(, qµ(q / /(q, where the sum is over all ivisors of for which gc(, q t. Both results rely o a ew form of Möbius iversio. Key wors. irreucible polyomial, trace, fiite fiel, Lyo wor, Möbius iversio MS subject classificatios. 05T06, 11T06 PII. S Itrouctio. The trace of a egree polyomial f(x over GF (q is the coefficiet of x 1. It is well kow that the umber of egree irreucible polyomials over GF (q is give by (1.1 I q ( = 1 µ(q /, where µ( is the Möbius fuctio. Less well kow is the formula (1.2 I 2 (, 1 = 1 2 o µ(2 /, which is the umber of egree irreucible polyomials over GF (2 with trace 1 (this ca be iferre from results i Jugickel [3, sectio 2.7]. Oe purpose of this paper is to refie (1.1 a (1.2 by eumeratig the irreucible egree polyomials over GF (q with a give trace. Carlitz [1] also solve this problem, arrivig via a ifferet techique at a expressio that is ifferet but equivalet to the oe give below. Our versio of the result is state i Theorem 1.1. Theorem 1.1. Let q be a power of prime p. The umber of irreucible polyomials of egree > 0 over GF (q with a give ozero trace t is (1.3 I q (, t = 1 q µ(q /. p Receive by the eitors Jauary 10, 2000; accepte for publicatio (i revise form Jauary 2, 2001; publishe electroically pril 3, Departmet of Computer Sciece, Uiversity of Victoria, EOW 348, 3800 Fierty Roa, P.O. Box 3055 MS 7209, Victoria, BC V8W 3P6, Caaa (fruskey@csr.uvic.ca, jsawaa@csr.uvic.ca. The research of these authors was supporte i part by NSERC. Departmet of Mathematics a Statistics, Uiversity of Victoria, Clearihue Builig, Room D268, 3800 Fierty Roa, Victoria, BC V8P 5C2, Caaa (crmiers@math.uvic.ca. 240

2 POLYNOMILS ND LYNDON WORDS WITH GIVEN TRCE 241 Note that the expressio o the right-ha sie of (1.3 is iepeet of t a that I q (, 0 ca be obtaie by subtractig I q (, 0 = I q ( (q 1I q (, 1. Lyo wor is the lexicographically smallest rotatio of a aperioic strig. If L q ( eotes the umber of q-ary Lyo wors of legth, the it is well kow that L q ( = I q (. The trace of a Lyo wor is the sum of its characters mo q. Let L q (, t eote the umber of Lyo wors of trace t. The seco purpose of this paper is to obtai a explicit formula for L q (, t. This result is state i Theorem 1.2. Theorem 1.2. For all itegers > 0, q > 1, a t {0, 1,..., q 1}, L q (, t = 1 q gc(,q t gc(, qµ(q /. Note that I q (, t = L q (, s wheever t 0 a gc(, s = 1. I orer to prove Theorems 1.1 a 1.2 we ee a ew form of Möbius iversio. This is presete i the ext sectio. 2. geeralize Möbius iversio formula. The efiig property of the Möbius fuctios is (2.1 µ( = [ = 1], where [P ] for propositio P represets the Iversoia covetio : [P ] has value 1 if P is true a value 0 if P is false (see [4, p. 24]. Defiitio 2.1. Let R be a set, N = {1, 2, 3,...}, a let {X(, t} t R, N be a family of subsets of R. We say that {X(, t} t R, N is recombiat if (i X(1, t = {t} for all t R a (ii {e X(, e : e X(, t} = {e X(, t} for all, N, t R. Theorem 2.2. Let {X(, t} t R, N be a recombiat family of subsets of R. Let : N R C a B : N R C be fuctios, where C is a commutative rig with ietity. The (, t = ( B, e e X(,t for all N a t R if a oly if B(, t = µ( e X(,t (, e for all N a t R. Proof. Cosier the sum, call it S, o the right-ha sie of the first equatio S = ( B, e e X(,t = ( µ(, e e X(,t (/ e X(,e = ( µ(, e. e X(,t e X(,e

3 242 F. RUSKEY, C. R. MIERS, ND J. SWD Now substitute f = a use recombiatio to get S = ( f [f = ]µ f e X(,t = ( f ( µ f, e f f e X(f,t = ( ( f f, e µ f e X(f,t f = ( f, e [f = 1] f e X(f,t = (, t. e X(,e ( f, e Verificatio i the other irectio is similar a is omitte. Lemma 2.3. Let N a e, t be members of a aitive mooi R. The sets {e : e = t} form a recombiat family. Proof. Here e meas e + e + + e ( terms. Suppose that e = t a e = e. Clearly, e = t. Coversely, if e = t, the e is equal to some elemet of R, call it e. The e = e a e = t. Corollary 2.4. For a fixe prime power q, the sets X q (, t = {e GF (q : e = t} form a recombiat family of subsets of GF (q. Corollary 2.5. For a fixe iteger q, the sets X q (, t = {e Z q : e t(q} form a recombiat family of subsets of Z q, where Z q are the itegers mo q. 3. Irreucible polyomials with give trace. I this sectio, the irreucible polyomials with a give trace are coute. We begi by itroucig some otatio that will be use i the remaier of the paper. We use Jugickel [3] as a referece for termiology a basic results from fiite fiel theory. The trace of a elemet β GF (q over GF (q is eote T r(β a is give by T r(β = β + β q + β q2 + + β q 1. If β GF (q a is the smallest positive iteger for which β q = 1, the f(x is the miimal polyomial of β, eote Mi(β, where f(x = (x β(x β q (x β q 1. The value of must be a ivisor of. Let Irr q (, t eote the set of all moic irreucible polyomials over GF (q of egree a trace t. By a Irr q (, t we eote the multiset cosistig of a copies of Irr q (, t. Classic results of fiite fiel theory imply the followig equality of multisets: (3.1 β GF(q {Mi(β} = Irr q ( = ( Irr q, where Irr q ( is the set of moic irreucible polyomials of egree over GF (q. From (3.1 it is easy to erive (1.1 via a staar applicatio of Möbius iversio.

4 (3.2 (3.3 (3.4 (3.5 POLYNOMILS ND LYNDON WORDS WITH GIVEN TRCE 243 Now we restrict the equality (3.1 to trace t fiel elemets to obtai β GF(q T r(β=t {Mi(β} = = = = { ( f Irr q { ( f Irr q e=t e=t { ( f Irr q } : T r(f = t {f Irr q (, e }. } : T r(f = t } : T r(f = e Note that the equatio e = t is askig whether the -fol sum of e GF (q is equal to t GF (q. We use the otatio GF (q, t for the set of elemets i GF (q with trace t, for t = 0, 1,..., q 1, where q = p m a p is prime. Cosier the map ρ that ses α to α + γ, where γ GF (q has trace 1. We claim that ρ(gf (q, t = GF (q, t + 1, a so the umber of elemets is the same for each trace value. Thus Takig carialities i (3.5 gives GF (q, t = q 1. q 1 = e=t ( I q, e. From Theorem 2.2 a Corollary 2.4, we obtai I q (, t = 1 q µ(q /. e=t The equatio e = t where is a iteger a e, t GF (q has a uique solutio e if t 0 a p. If t = 0, the there is oe solutio e = 0 if p a there are q solutios if p. Thus, if t 0, the I q (, t = 1 q µ(q /, p thereby provig Theorem 1.1. Otherwise, if t = 0, the I q (, 0 = I q (, µ(q /. 4. Lyo wors with give trace. If a = a 1 a 2 a is a wor, the we efie its trace mo q, T r q (a, to be a i mo q. Let L q (, t eote the umber of q-ary Lyo wors of legth a trace t mo q. Note that ay q-ary strig of legth ca be expresse as the cocateatio of copies of the rotatio of some Lyo wor of legth / for some. Note further that there are precisely q 1 p

5 244 F. RUSKEY, C. R. MIERS, ND J. SWD wors of legth with trace t because ay wor of legth 1 ca have a fial th character appee i oly oe way to have trace t. It therefore follows that q 1 = ( (4.1 L q, e. e t(q This ca be solve usig Theorem 2.2 a Corollary 2.5 to yiel L q (, t = µ( q / 1. e t(q Hece (4.2 L q (, t = 1 q gc(q, t gc(q, µ(q /. Equatio (4.2 is true because e t(q has a solutio if a oly if gc(, q t. If a solutio exists, the it has precisely gc(, q solutios (e.g., [2, Corollary 33.22, p. 821]. This proves Theorem 1.2. We coul also cosier the more geeral questio of computig L q,r (, t, the umber of q-ary Lyo wors with trace mo r, a erive similar but more complicate formulae. If M q (, t is the umber of q-ary legth strigs whose characters sum to t, the clearly M q (1, t = [0 t < q ] a for > 1 M q (, t = q M q ( 1, t i. i=0 If T q,r (, t eotes the umber of q-ary legth strigs with trace mo r equal to t, the T q,r (, t = M q (, s. Usig the same approach as before s t(r L q,r (, t = 1 µ( e t(r T q,r (, e. The equatio for L q,r (, t seems to prouce o particularly ice formulae, except i the case see previously where q = r or if q = 2. Whe q = 2, M 2 (, t = ( t a T 2,r (, t = (. s s t(r However, i this case there is alreay a well-kow formula for the umber of Lyo wors with k 1 s, amely, P 2 (, k = 1 ( / µ(, k/ gc(,k from which we obtai L 2,r (, t = s t(2 P 2(, s.

6 POLYNOMILS ND LYNDON WORDS WITH GIVEN TRCE Fial remarks. Our geeralize Möbius iversio theorem ca be extee to a Möbius iversio theorem o posets. Backgrou material o Möbius iversio o posets may be fou i Staley [5]. We state here the moifie efiitio of recombiat a the iversio theorem but omit the proof. Defiitio 5.1. Let P be a poset, let R be a set, a let {X(y, x, t} x,y P,y x,t R be a family of subsets of R. The family {X(y, x, t} x,y P,y x,t R is recombiat if (i X(x, x, t = {t} for all t R a (ii {e X(z, y, e : e X(y, x, t} = {e X(z, x, t} for all z y x P, t R. We ote that if P is the ivisor lattice a R is a aitive mooi, the the collectio {X(x, y, t} x,y P,x y,t R where X(x, y, t = {e R : (y/xe = t} is recombiat, as per Lemma 2.3. Theorem 5.2. Let P be a poset, let R be a set, a let {X(y, x, t} x,y P,y x,t R be a recombiat family. Let : P R C, a B : P R C, be fuctios where C is a commutative rig with ietity. The (x, t = B(y, e y x for all x P a t R if a oly if B(x, t = y x µ(y, x e X(y,x,t e X(y,x,t (y, e for all x P a t R. (Here µ(y, x is the Möbius fuctio of the poset P. Tables of the umbers I q (, t a L q (, t for small values of q a may be fou o Frak Ruskey s combiatorial object server (COS at cos/if/{lyo.html,irreucible.html}. They also appear i Neil Sloae s o-lie ecyclopeia of iteger sequeces (at jas/sequeces/ as I 2 (, 0 = L 2 (, 0 = , I 2 (, 1 = L 2 (, 1 = , I 3 (, 0 = L 3 (, 0 = , I 3 (, 1 = L 3 (, 1 = , L 4 (, 0 = , I 4 (, 1 = L 4 (, 1 = , L 5 (, 0 = , I 5 (, 1 = L 5 (, 1 = , L 6 (, 0 = , L 6 (, 1 = , L 6 (, 2 = , L 6 (, 3 = ckowlegmet. The authors wish to thak aro Gulliver for helpful iscussios regarig this paper. REFERENCES [1] L. Carlitz, theorem of Dickso o irreucible polyomials, Proc. mer. Math. Soc., 3 (1952, pp [2] T.H. Corme, C.E. Leiserso, a R.L. Rivest, Itrouctio to lgorithms, McGraw-Hill, New York, [3] D. Jugickel, Fiite Fiels: Structure a arithmetics, B.I. Wisseschaftsverlag, Maheim, Germay, [4] D.E. Kuth, R.L. Graham, a O. Patashik, Cocrete Mathematics, iso-wesley, Reaig, M, [5] R.P. Staley, Eumerative Combiatorics, Vol. I, Cambrige Uiversity Press, Cambrige, UK, 1997.