ON CYCLOTOMIC POLYNOMIALS WITH ±1 COEFFICIENTS

Transcription

1 ON CYCLOTOMIC POLYNOMIALS WITH ±1 COEFFICIENTS PETER BORWEIN AND KWOK-KWONG STEPHEN CHOI Abstract. We characterze all cyclotomc polynomals of even egree wth coeffcents restrcte to the set {+1, 1}. In ths context a cyclotomc polynomal s any monc polynomal wth nteger coeffcents an all roots of moulus 1. Inter ala we characterze all cyclotomc polynomals wth o coeffcents. The characterzaton s as follows. A polynomal P (x) wth coeffcents ±1 of even egree N 1 s cyclotomc f an only f P (x) = ±Φ p1 (±x)φ p2 (±x p1 ) Φ pr (±x p1p2 pr 1 ), where N = p 1 p 2 p r an the p are prmes, not necessarly stnct. Here Φ p (x) := x p 1 x 1 s the pth cyclotomc polynomal. We conjecture that ths characterzaton also hols for polynomals of o egree wth ±1 coeffcents. Ths conjecture s base on substantal computaton plus a number of specal cases. Central to ths paper s a careful analyss of the effect of Graeffe s root squarng algorthm on cyclotomc polynomals. Keywors: Cyclotomc polynomals, Mahler measure, Lttlewoo polynomals AMS (1991) subject classfcaton: Prmary 11R09, Seconary 11Y99 Research of P. Borwen s supporte, n part, by NSERC of Canaa. K.K. Cho s a Pacfc Insttute of Mathematcs Postoctoral Fellow an the Insttute s support s gratefully acknowlege. 1

2 2 PETER BORWEIN AND KWOK-KWONG STEPHEN CHOI 1. Introucton. We are ntereste n stuyng polynomals wth coeffcents restrcte to the set {+1, 1}. Ths partcular set of polynomals has rawn much attenton an there are a number of ffcult ol questons concernng t. Lttlewoo rase a number of these questons an so we call these polynomals Lttlewoo polynomals an enote them by L [6]. A Lttlewoo polynomal of egree n has L 2 norm on the unt crcle equal to n + 1. Many of the questons rase concern comparng the behavor of these polynomals n other norms to the L 2 norm. One of the oler an more ntrgung of these asks whether such polynomals can be flat. Specfcally, o there exst two postve constants C 1 an C 2 so that for each n there s Lttlewoo polynomal of egree n wth C 1 n + 1 < p(z) < C2 n + 1 for each z of moulus 1. Ths problem whch has been open for more than forty years s scusse n [1] where there s an extensve bblography. The upper boun s satsfe by the so calle Run Shapro polynomals. It s stll open as to whether there s a sequence satsfyng just the lower boun (ths problem has been calle one of the very harest problems n combnatoral optmzaton ). The sze of the L p norm of Lttlewoo polynomals has been stue from a number of ponts of vew. The problem of mnmzng the L 4 norm (or equvalently of maxmzng the so calle mert factor ) has also attracte a lot of attenton. In partcular can Lttlewoo polynomals of egree n have L 4 norm asymptotcally close to n + 1. Ths too s stll open an s scusse n [1]. Mahler [7] rase the queston of maxmzng the Mahler measure of Lttlewoo polynomals. The Mahler measure s just the L 0 norm on the crcle an one woul expect ths to be closely relate to the mnmzaton problem for the L 4 norm above. Of course the mnmum possble Mahler measure for a Lttlewoo polynomal s 1 an ths s acheve by any cyclotomc polynomal. In ths paper a cyclotomc polynomal s any monc polynomal wth nteger coeffcents an all roots of moulus 1. Whle Φ n (x) enotes the nth rreucble cyclotomc polynomal (the mnmum polynomal of a prmtve nth root of unty). Ths paper aresses the queston of characterzng the cyclotomc Lttlewoo polynomals of even egree. Specfcally we show that a polynomal P (x) wth coeffcents ±1 of even egree N 1 s cyclotomc f an only f P (x) = ±Φ p1 (±x)φ p2 (±x p1 ) Φ pr (±x p1p2 pr 1 ), where N = p 1 p 2 p r an the p are prmes (not necessarly stnct). The f part s obvous snce Φ p (x) has coeffcents ±1. We also gve an explct formula for the number of such polynomals. Ths analyss s base on a careful treatment of Graeffe s root squarng algorthm. It transpres that all cyclotomc Lttlewoo polynomals of a fxe egree have the same fxe pont on teratng Graeffe s root squarng algorthm. Ths allows us to also characterze all cyclotomc polynomals wth o coeffcents. Substantal computatons, as well as a number of specal cases, lea us to conjecture that the above characterzaton of cyclotomc Lttlewoo polynomals of even egree also hols for o egree. One of the cases we can hanle s when N s a power of 2.

3 ON CYCLOTOMIC POLYNOMIALS WITH ±1 COEFFICIENTS 3 It s worth commentng on the expermental aspects of ths paper. (As s perhaps usual, much of ths s carefully erase n the fnal exposton). It s really the observaton that the cyclotomc Lttlewoo polynomals can be explctly constructe essentally by nvertng Graeffe s root squarng algorthm that s crtcal. Ths allows for computaton over all cyclotomc Lttlewoos up to egree several hunre (wth exhaustve search falng far earler). A constructon whch s of nterest n tself. Inee t was these calculatons that allowe for the conjectures of the paper an suggeste the route to some of the results. The paper s organze as follows. Secton 2 examnes cyclotomc polynomals wth o coeffcents. Secton 3 looks at cyclotomc Lttlewoo polynomals wth a complete analyss of the even egree case. The last secton presents some numercal evence an other evence to support the conjecture n the o case behaves lke the even case. 2. Cyclotomc Polynomals wth O Coeffcents. In ths secton, we scuss the factorzaton of cyclotomc polynomals wth o coeffcents as a prouct of rreucble cyclotomc polynomals. To o ths, we frst conser the factorzaton over Z p [x] where p s a prme number. The most useful case s p = 2 because every Lttlewoo polynomal reuces to the Drchlet kernel 1 + x + + x N 1 n Z 2 [x]. In Z p [x], Φ n (x) s no longer rreucble n general but Φ n (x) an Φ m (x) are stll relatvely prme to each other. Here, as before, Φ n (x) s the nth rreucble cyclotomc polynomal. Lemma 2.1. Suppose n an m are stnct postve ntegers relatvely prme to p. Then Φ n (x) an Φ m (x) are relatvely prme n Z p [x]. Proof. Suppose e an f are the smallest postve ntegers such that p e 1 (mo n) an p f 1 (mo m). Let F p k be the fel of orer p k. Then F p e contans exactly φ(n) elements of orer n an over Z p, Φ n (x) s a prouct of φ(n)/e rreucble factors of egree e an each rreucble factor s a mnmal polynomal for an element n F p e of orer n over Z p, see [5]. So Φ n (x) an Φ m (x) cannot have a common factor n Z p [x] snce ther rreucble factors are mnmal polynomals of fferent orers. Ths proves our lemma. The followng lemma tells whch Φ m (x) can possbly be factors of polynomals wth o coeffcents. Lemma 2.2. Suppose P (x) s a polynomal wth o coeffcents of egree N 1. If Φ m (x) ves P (x), then m ves 2N. Proof. Snce Φ m (x) ves P (x), so Φ m (x) also ves P (x) n Z 2 [x]. However, n Z 2 [x], P (x) equals to 1 + x + + x N 1 an can be factore as (2.1) P (x) = Φ 1 (x) 1 M Φ 2t (x), where N = 2 t M, t 0 an M s o. In vew of Lemma 2.1, Φ 1 (x) an Φ 2 (x) are relatvely prme n Z 2 [x] f 1 an 2 are stnct o ntegers. So f m s o, then Φ m (x) s a factor n the rght han se of (2.1) an hence m = for some

4 4 PETER BORWEIN AND KWOK-KWONG STEPHEN CHOI M. On the other han, f m s even an m = 2 l m where l 1 an m s o, then Φ m (x) = Φ 2m (x 2l 1 ) = Φ m (x 2l 1 ) = Φ m (x) 2l 1 n Z 2 [x]. Thus n vew of (2.1), we must have m = for M an l t + 1. Hence n both cases, we have m ves 2N. In vew of Lemma 2.2, every cyclotomc polynomal, P (x), wth o coeffcents of egree N 1 can be wrtten as (2.2) P (x) = Φ e() (x) 2N where e() are non-negatve ntegers. For each prme p let T p be the operator efne over all monc polynomals n Z[x] by N T p [P (x)] := (x α p ) for every P (x) = N (x α ) n Z[x]. By Newton s enttes (see [3], p.5), T p [P (x)] s also a monc polynomal n Z[x]. We exten T p to be efne over the quotent of two monc polynomals n Z[x] by T p [(P/Q)(x)] := T p [P (x)]/t P [Q(x)]. Ths operator obvously takes a polynomal to the polynomal whose roots are the pth powers of the roots of P. Also we let M p be the natural projecton from Z[x] onto Z p [x]. So, M p [P (x)] = P (x) (mo p). Lemma 2.3. Let n be a postve nteger relatvely prme to p an 2. Then we have () T p [Φ n (x)] = Φ n (x), () T p [Φ pn (x)] = Φ n (x) p 1, () T p [Φ p n(x)] = Φ p 1 n(x) p. Proof. () s trval because f (n, p) = 1 then T p just permutes the roots of Φ n (x). To prove () an (), we conser N T p [P (x p )] = T p [ (x p α )] j=1 N p = T p [ (x e 2πl = N j=1 l=1 j=1 l=1 = P (x) p. p (x α ) 1 p α p Thus () an () follow from (), Φ pn (x) = Φn(xp ) Φ n(x) an Φ p n(x) = Φ p 1 n(x p ) (see [8], 5.8). )]

5 ON CYCLOTOMIC POLYNOMIALS WITH ±1 COEFFICIENTS 5 When P (x) s cyclotomc, the terates T n p [P (x)] converge n a fnte number of steps to a fxe pont of T p an we efne ths to be the fxe pont of P (x) wth respect to T p. Lemma 2.4. If P (x) s a monc cyclotomc polynomal n Z[x], then (2.3) M p [T p [P (x)]] = M p [P (x)], n Z p [x], where M p s the above natural projecton. Proof. Snce both T p an M p are multplcatve, t suffces to conser the prmtve cyclotomc polynomals Φ n (x). Let n be an nteger relatvely prme to p. Then (2.3) s true for P (x) = Φ n (x) by () of Lemma 2.3. For P (x) = Φ pn (x), we have by () of Lemma 2.3. However, M p [T p [Φ pn (x)]] = M p [Φ n (x) p 1 ] = M p [Φ n (x)] p 1. M p [Φ pn (x)] = M p[φ n (x p )] M p [Φ n (x)] = M p[φ n ](x p ) M p [Φ n (x)] = M p [Φ n (x)] p 1, n Z p [x]. Ths proves that (2.3) s also true for P (x) = Φ pn (x). Fnally, f P (x) = Φ p n(x) then M p [T p [Φ p n(x)]] = M p [Φ p 1 n(x) p ] = M p [Φ p 1 n(x p )] = M p [Φ p n(x)] by () of Lemma 2.3. Ths completes the proof of our lemma. Lemma 2.4 shows that f T p [P (x)] = T p [Q(x)] then M p [P (x)] = M p [Q(x)]. The next result shows that the converse s also true. Theorem 2.5. P (x) an Q(x) are monc cyclotomc polynomals n Z[x] an M p [P (x)] = M p [Q(x)] n Z p [x] f an only f both P (x) an Q(x) have the same fxe pont wth respect to teraton of T p. Proof. Suppose P (x) = Φ e() (x)φ e(p) p (x) Φ e(pt ) p t (x) D an Q(x) = Φ e() (x)φ e(p) p (x) Φ e(pt ) p t (x) D where t, e(j), e(j) 0 an D s a set of postve ntegers relatvely prme to p. Then usng ()-() of Lemma 2.3, we have for l t (2.4) T l p[p (x)] = D Φ (x) f() an T l p[q(x)] = D Φ (x) f() where t f() = e() + (p 1) p j 1 e(p j ) j=1

6 6 PETER BORWEIN AND KWOK-KWONG STEPHEN CHOI an From Lemma 2.4, we have f() = e() + (p 1) t p j 1 e(p j ). j=1 M p [T l p[p (x)]] = M p [P (x)] = M p [Q(x)] = M p [T l p[q(x)]], for any l t. From ths an (2.4), D M p [Φ (x)] f() = D M p [Φ (x)] f(). However, wth Lemma 2.1, M p [Φ (x)] an M p [Φ (x)] are relatvely prme f. So we must have f() = f() for all D an hence from (2.4), P (x) an Q(x) have the same fxe pont wth respect to T p. From Theorem 2.5, we can characterze the monc cyclotomc polynomals by ther mages n Z p [x] uner the projecton M p. They all have the same fxe pont uner T p. In partcular, when p = 2 we have Corollary 2.6. All monc cyclotomc polynomals wth o coeffcents of the egree N 1 have the same fxe pont uner teraton of T 2. Specfcally, f N = 2 t M where t 0 an (2, M) = 1 then the fxe pont occurs at the t + 1-th step of the teraton an equals (x M 1) 2t (x 1) 1. Proof. The frst part follows rectly from our Theorem 2.5 an the fact that M 2 [P (x)] = 1 + x + + x N 1 n Z 2 [x] f P (x) s a monc polynomals wth o coeffcents of egree N 1. If N = 2 t M, then from (2.2), P (x) = M Φ e() (x)φ e(2) 2 (x) Φ e(2t+1 ) 2 t+1 (x). Over Z 2 [x], 1 + x + + x N 1 = Φ 1 (x) 1 M Φ 2t (x), so t+1 (2.5) f() = e() e(2 ) = Therefore, from (2.5) an Lemma 2.3, T t+1 2 [P (x)] = M { 2 t for M, > 1, 2 t 1 for = 1. Φ f() (x) = Φ 1 (x) 1 Φ 2t (x) = (x M 1) 2t (x 1) 1. M Corollary 2.6, when N s o (t = 0) shows that T 2 [P (x)] equals to 1 + x + + x N 1 for all cyclotomc polynomals wth o coeffcents an from (2.2) an (2.5), we have the followng characterzaton of cyclotomc polynomals wth o coeffcents.

7 ON CYCLOTOMIC POLYNOMIALS WITH ±1 COEFFICIENTS 7 Corollary 2.7. Let N = 2 t M wth t 0 an (2, M) = 1. A polynomal, P (x), wth o coeffcents of egree N 1 s cyclotomc f an only f P (x) = M Φ e() (x)φ e(2) 2 (x) Φ e(2t+1 ) 2 t+1 (x), an the e() satsfy the conton (2.5). Furthermore, f N s o, then any polynomal, P (x), wth o coeffcents of even egree N 1 s cyclotomc f an only f P (x) = N,>1 where the e() are non-negatve ntegers. Φ e() (±x) Corollary 2.7 allows us to compute the number of cyclotomc polynomals wth o coeffcents. Let B(n) be the number of parttons of n nto a sum of terms of the sequence {1, 1, 2, 4, 8, 16, }. Then B(n) has generatng functon F (x) = (1 x) 1 It follows from (2.5) an Corollary 2.7 that k=0 (1 x 2k ) 1. Corollary 2.8. Let N = 2 t M wth t 0 an (2, M) = 1. The number of cyclotomc polynomals wth o coeffcents of egree N 1 s (2.6) C(N) = B(2 t ) (M) 1 B(2 t 1) where (M) enotes the number of vsors of M. Furthermore, (2.7) log C(N) ( t2 2 log 2)((M) 1) + (log(2t 1)) 2. log 4 Proof. Formula (2.6) follows from (2.5) an Corollary 2.7. To prove (2.7), we use e Brujn s asymptotc estmaton for B(n) n [4]: Now (2.7) follows from ths an (2.6). B(n) exp((log n) 2 / log 4). 3. Cyclotomc Lttlewoo Polynomals. We now specalze the scusson to the case where the coeffcents are all plus one or mnus one. One natural way to bul up Lttlewoo polynomal of hgher egree s as follows: f P 1 (x) an P 2 (x) are Lttlewoo polynomals an P 1 (x) s of egree N 1 then P 1 (x)p 2 (x N ) s a Lttlewoo polynomals of hgher egree. In ths secton, we are gong to show that ths s also the only way to prouce cyclotomc Lttlewoo polynomals, at least, for even egree. To prove ths, t s equvalent to show that the coeffcents of P (x) are peroc n the sense that f P (x) = N 1 a nx n, then there s a pero such that a l+n = a l for all 1 n 1 an 0 l N/ 1. Ths s our Theorem 3.3 below.

8 8 PETER BORWEIN AND KWOK-KWONG STEPHEN CHOI Suppose P (x) = N 1 a nx n s a cyclotomc polynomal n L an let S k be the sum of k-th power of all the roots of P (x). Snce P (x) s cyclotomc, we have x N 1 P (1/x) = ±P (x). Thus t follows from Newton s enttes that (3.1) S k + a 1 S k a k 1 S 1 + ka k = 0 for k N 2. We may further assume that a 0 = a 1 = 1 by replacng P (x) by P (x) or P ( x) f necessary. We now let a 0 = a 1 = = a 1 = 1 an a = 1, for some nteger 2. From (3.1), we have We clam that S 1 = a 1 = 1. (3.2) S 1 = S 2 = = S 1 = 1 an S = 2 1. Suppose S 1 = = S j = 1 for j < 1. Then from (3.1) agan, S j+1 = a 1 S j a j S 1 (j + 1)a j+1 = j (j + 1) = 1. So S 1 = = S 1 = 1. Smlarly, from (3.1) S = a 1 S 1 a 1 S 1 a = 2 1. Lemma 3.1. Let 2 k N 1 1 an suppose a l+n = a l for 1 n 1 an 0 l k 2. Then we have (3.3) k 2 { } a l S(k l)+j+1 S (k l 1)+j+1 + (k + j + 1)(ak+j+1 a k+j ) l=0 for 0 j a (k 1)+n (S +j n+1 S +j n ) = 0 Proof. Suppose 0 j 2. From (3.1) an (3.2) we have (3.4) Smlarly, 0 = = k 2 (3.5) 0 = l=0 j 1 a l+n S (k l)+j n + a k+n S j n + (k + j)a k+j k 1 1 l=0 k 2 1 a l l=0 a l 1 1 S (k l)+j n + a (k 1)+n S +j n j a k+n + (k + j + 1)a k+j. 1 S (k l)+j n+1 + a (k 1)+n S +j n+1 j a k+n + (k + j + 1)a k+j+1.

9 ON CYCLOTOMIC POLYNOMIALS WITH ±1 COEFFICIENTS 9 Hence, on subtractng (3.5) from (3.4), we have k 2 { } 0 = a l S(k l)+j+1 S (k l 1)+j+1 + (k + j + 1)(ak+j+1 a k+j ) l=0 Ths proves (3.3). 1 + a (k 1)+n (S +j n+1 S +j n ). Lemma 3.2. Let 0 k N 1 1. Suppose a l+n = a l for 1 n 1 an 0 l k. Then (3.6) S l+n = 1 for 1 n 1 an 0 l k. Proof. We prove ths by nucton on k. We have prove that (3.6) s true for k = 0. Suppose (3.6) s true for k 1. Then for any 0 j 2, 1 0 = a 0 {S k+j+1 S (k 1)+j+1 } + a (k 1) (S +j n+1 S +j n ) = S k+j a (k 1) (S +j+1 S j+1 ) = S k+j+1 + 1, by (3.3). Hence S k+j+1 = 1 for 0 j 2. Theorem 3.3. Suppose N s o. If a 0 = a 1 = = a 1 = 1 an a = 1, then for 1 n 1 an 0 l N 1. a l+n = a l Proof. We frst show that S 2k = 1 for 1 k N 1 an hence s o because S = 2 1. Snce N s o, from Corollary 2.7, P (x) = N Φ e() (x)φ e(2) 2 (x) where e() + e(2) = 1 f > 1 an e(1) = e(2) = 0. If 1 k N 1, then S 2k = N (e()c (2k) + e(2)c 2 (2k)) = N(e() + e(2))c (k) (3.7) = N C (k) C 1 (k) = 1 where the Ramanujan sum, C (k), s the sum of the kth powers of the prmtve th roots of unty an hence N C (k) s the sum of kth powers of the roots of N Φ (x) = x N 1 whch s equal to zero when 1 k N 1. We then procee our proof by usng nucton on k. Suppose a l+n = a l

10 10 PETER BORWEIN AND KWOK-KWONG STEPHEN CHOI for 1 n 1 an 0 l k 1 where 1 k N 1 1. From Lemmas 3.1 an 3.2 we have (3.8) S k+j (k + j + 1)(a k+j+1 a k+j ) = 0 an hence from (3.3) agan 0 = a 0 (S (k+1)+j+1 S k+j+1 ) + a (S k+j+1 S (k 1)+j+1 ) + a k+j (S +1 S ) (3.9) +a k+j+1 (S S 1 ) + ((k + 1) + j + 1)(a (k+1)+j+1 a (k+1)+j ) = (S (k+1)+j+1 2S k+j+1 1) + 2(a k+j+1 a k+j ) +((k + 1) + j + 1)(a (k+1)+j+1 a (k+1)+j ) = S (k+1)+j ((k + 1) + j + 1)(a k+j+1 a k+j ) +((k + 1) + j + 1)(a (k+1)+j+1 a (k+1)+j ) for 0 j 2. Suppose k s even. Then n vew of (3.7), S k+j+1 = 1 f j s o an S (k+1)+j+1 = 1 f j s even. So from (3.8) an (3.9), we have for j = 1, 3,, 2 an a k+j+1 = a k+j (3.10) 2(a k+j+1 a k+j ) = a (k+1)+j+1 a (k+1)+j for j = 0, 2,, 3. However, snce the a s are +1 or 1, (3.10) mples that a k+j+1 = a k+j an a (k+1)+j+1 = a (k+1)+j for j = 0, 2,, 3. Hence a k+n = a k for n = 1, 2,, 1. The case k s o can be prove n the same way. Theorem 3.4. Suppose N s o. A Lttlewoo polynomal, P (x), of egree N 1 s cyclotomc f an only f (3.11) P (x) = ±Φ p1 (±x)φ p2 (±x p1 ) Φ pr (±x p1p2 pr 1 ), where N = p 1 p 2 p r an the p are prmes, not necessarly stnct. Proof. It s clear that f P (x) s n the form of (3.11), then P (x) s a cyclotomc Lttlewoo polynomal. Conversely suppose P (x) s a cyclotomc Lttlewoo polynomal. As before we may assume that a 0 = a 1 = = a 1 = 1 an a = 1. Then we are gong to prove our result by nucton on N. From Theorem 3.3, we have P (x) = P 1 (x)p 2 (x ) where P 1 (x) = 1+x+ +x 1 an P 2 (x) s a cyclotomc Lttlewoo polynomal of egree < N 1. By nucton, P 1 (x) an P 2 (x) are n the form of (3.11) an hence so s P (x) because the egree of P 1 (x) s 1. Ths completes the proof of our theorem. Corollary 3.5. Suppose N s o. Then P (x) s cyclotomc n L of egree N 1 f an only f t x N + ( 1) ɛ+ P (x) = ± x N 1 + ( 1) ɛ+ where ɛ = 0 or 1, N 0 = 1, N t = N an N 1 s a proper vsor of N for = 1, 2,, t.

11 ON CYCLOTOMIC POLYNOMIALS WITH ±1 COEFFICIENTS 11 Proof. Wthout loss of generalty, we may assume that P (x) = 1 + x + a 2 x 2 +. So from Theorem 3.4, P (x) s cyclotomc n L f an only f P (x) = Φ p1 (x)φ p2 (±x p1 ) Φ pr (±x p1 pr 1 ) (3.12) = Φ p1 (x) Φ pn1 (x p1 pn 1 1 )Φ pn1 +1( x p1 pn 1 ) Φpn2 ( x p1 pn 2 1 ) Φ pnt 1 +1 (( 1)t 1 x p1 pn t 1 ) Φpnt (( 1) t 1 x p1 pn t 1 ) where N = p 1 p nt. Snce Φ p (x) = xp 1 x 1, (3.12) becomes t x N + ( 1) P (x) = x N 1 + ( 1), where N 0 = 1 an N = p 1 p n for = 1,, t. Ths proves our corollary. Usng Corollary 3.5, we can count the number of cyclotomc Lttlewoo polynomals of gven even egree. For any postve ntegers N an t, efne r(n, t) := #{(N 1, N 2,, N t ) : N 1 N 2 N t, 1 < N 1 < N 2 < < N t = N}; an for 1, (3.13) (N) := n N 1 (n) where 0 (N) = 1. Lemma 3.6. For l, t 0 an p prme, we have ( ) l + t (3.14) t (p l ) =. t Proof. We prove the lemma by nucton on t. Equalty (3.14) s clearly true for t = 0 because 0 (N) = 1. We then suppose (3.14) s true for t 1 where t 1. Then t (p l ) = l l ( ) + t 1 t 1 (n) = t 1 (p ) =. t 1 n p l So t (p l ) s the coeffcent of x t 1 n =0 =0 (x + 1) t 1 + (x + 1) t + + (x + 1) l+t 1 { (x + 1) = (x + 1) t 1 l+1 } 1 x = (x + 1)l+t (x + 1) t 1. x Hence ) t (p l ) s the coeffcent of x t n (x + 1) l+t (x + 1) t 1. Therefore, t (p l ) =. ( l+t t Snce t (N) s a multplcatve functon of N, we have Corollary 3.7. If N = p r1 1 prs s where r 1 an p are stnct prmes, then s ( ) r + t t (N) =. t

12 12 PETER BORWEIN AND KWOK-KWONG STEPHEN CHOI Lemma 3.8. For any postve ntegers N an t, we have { 0 f N = 1, (3.15) r(n, t) := t ( 1)t ( t ) 1 (N) f N > 1. Proof. We agan prove by nucton on t. It s clear from the efnton that r(1, t) = 0 an r(n, 1) = 1 for any t, N 1. We then suppose N > 1 an (3.15) s true for t 1 where t 2. Then r(n, t) = N 1 N = N 1>1 N 1 N N>N 1>1 { t 1 r(n/n 1, t 1) r(n/n 1, t 1) = ( ) } t 1 ( 1) t 1 1 (N/N 1 ) N 1 N t 1 ( ) t 1 ( 1) t 1 { 1 (1) + 1 (N)} t ( ) t 1 = ( 1) t 1 (N) 1 =2 t 1 ( ) t 1 ( 1) t 1 1 (N) + ( 1) t 1 t ( ) t = ( 1) t 1 (N) from (3.13) an the fact that ( ) ( t t 1 ) ( = t ). Corollary 3.9. The number of cyclotomc polynomals n L of egree N 1 where N = p r1 1 prs s, r 1 an the p are stnct o prmes, s r 1+ +r s ( ) s ( ) 4 ( 1) j rk + j 1. j j 1 j=1 k=1 Proof. From Corollary 3.5, the number of cyclotomc polynomals n L of egree N 1 s r 1+ +r s 4 r(n, ). The corollary now follows from Corollary 3.7 an Lemma Cyclotomc Lttlewoo Polynomals of O Degree. We conjecture explctly that Theorem 3.4 also hols for polynomals of o egree.

13 ON CYCLOTOMIC POLYNOMIALS WITH ±1 COEFFICIENTS 13 Conjecture 4.1. A Lttlewoo polynomal, P (x), of egree N 1 s cyclotomc f an only f (4.1) P (x) = ±Φ p1 (±x)φ p2 (±x p1 ) Φ pr (±x p1p2 pr 1 ), where N = p 1 p 2 p r an the p are prmes, not necessarly stnct. We compute up to egree 210 (except for the case N 1 = 191). The computaton was base on computng all cyclotomc polynomals wth o coeffcents of a gven egree an then checkng whch were actually Lttlewoo an seeng that ths set matche the set generate by the conjecture. For example, for N 1 = 143 there are cyclotomc polynomals wth o coeffcents of whch 416 are Lttlewoo. For N 1 = 191 there are cyclotomc polynomals wth o coeffcents (whch was too bg for our program). We can generate all the cyclotomcs wth o coeffcents from Corollary 2.7 qute easly so the bulk of the work s nvolve n checkng whch ones have heght 1. The set n the conjecture computes very easly recursvely. Some specal cases also support the conjecture. Most notably the case where N s a power of 2. The proof s as follows. From Corollary 2.7, we have P (x) = Φ e(1) 1 (x)φ e(2) 2 (x) Φ e(2t+1 ) 2 t+1 (x). Agan, we assume a 0 = a 1 = 1. Snce Φ 1 (x)φ 2 (x) = x 2 1 an Φ 2 l(x) = Φ 2 (x 2l 1 ) for l 2, we have e(2) e(1) = 1 an hence P (x) = Φ 2 (x)q(x 2 ), for some cyclotomc Lttlewoo polynomal Q(x). Therefore, by nucton, P (x) satsfes (4.1). Acknowlegment The authors wsh to thank Professor D. Boy for hs avce an support concernng ths paper. References [1] P. Borwen, Some Ol Problems on Polynomals wth Integer Coeffcents (To appear). [2] P. Borwen, T. Erély & G. Kos Lttlewoo-type problems on [0, 1] Bull. Lonon Math. Soc. (to appear). [3] P. Borwen & T. Erély, Polynomals an Polynomals Inequaltes, GTM 161, Sprnger- Verlag, Berln, [4] N. G. e Brujn, On Mahler s partton problem, Inagatones Math.,10 (1948), [5] R. Ll & H. Neerreter, Fnte Fels, Ason-Wesley, Reang, [6] J.E. Lttlewoo Some Problems n Real an Complex Analyss, Heath Mathematcal Monographs, Lexngton, Mass [7] K. Mahler On two extremal propertes of polynomals, Illnos J. Math., 7 (1963), [8] T. W. Hungerfor, Algebra, GTM 73, Sprnger-Verlag, Berln, Department of Mathematcs an Statstcs, Smon Fraser Unversty, Burnaby, B.C., Canaa V5A 1S6 Department of Mathematcs, Unversty of Brtsh Columba, Vancouver, B.C., Canaa V6T 1Z2 an Department of Mathematcs an Statstcs, Smon Fraser Unversty, Burnaby, B.C., Canaa V5A 1S6