ON TESTING THE DIVISIBILITY OF LACUNARY POLYNOMIALS BY CYCLOTOMIC POLYNOMIALS Michael Filaseta* and Andrzej Schinzel 1. Introduction and the Main Theo

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1 ON TESTING THE DIVISIBILITY OF LACUNARY POLYNOMIALS BY CYCLOTOMIC POLYNOMIALS Michael Filaeta* and Andrzej Schinzel 1. Introduction and the Main Theorem Thi note decribe an algorithm for determining whether a given polynomial f(x) 2 Z[x] ha a cyclotomic divior. In particular, the algorithm work well when the number of non-zero term i mall compared to the degree of f(x). Thi work i baed on paper of H. B. Mann [3] and J. H. Conway and A. J. Jone [1]. For m a poitive integer, we define Φ m (x) to be the mth cyclotomic polynomial, and let m = e 2ßi=m. If m = k` where k and ` are relatively prime poitive integer, then it i not difficult to ee that Q( m ) = Q( k ; `). Letting ffi denote Euler' ffi-function, we oberve that [Q( k ) : Q] = ffi(k) and [Q( m ) : Q] = ffi(m) = ffi(k)ffi(`). It follow that [Q( m ) : Q( k )] = ffi(`) and that Φ`(x) i the minimal polynomial for ` over the field Q( k ). In particular, f1; `; 2`;:::; ffi(`) 1 ` g form a bai for Q( m ) over Q( k ). We will mainly be intereted in the cae when ` i a prime power. For integer a, b, and m with m > 0, we write a b (mod m) if mj(a b) and ue the notation a mod m to repreent the unique b a (mod m) uch that 0» b < m. For f(x), g(x), and w(x) in Q[x] with deg w(x) 1, we write f(x) g(x) (mod w(x)) if w(x)j(f(x) g(x)), and we ue the notation f(x) mod w(x) to denote the unique polynomial g(x) f(x) (mod w(x)) with either g(x) 0 or 0» deg g(x) < deg w(x). Theorem 1. Let f(x) 2 Z[x]. Let m, k, and ` be poitive integer, with k and ` relatively prime, m = k`, and Φ m (x) dividing f(x). Let (1) f(xy) mod Φ`(y) = ffi(`) 1 X Then Φ k (x) divide each a j (x) for 0» j» ffi(`) 1. a j (x)y j : Proof. Oberve that m = k` and k and ` being relatively prime imply that k ` i a primitive mth root of unity. Taking x = k and y = `, we deduce that X ffi(`) 1 a j ( k ) j` =0: On the other hand, f1; `; 2`;:::; ffi(`) 1 ` gi a bai for Q( m )over Q( k ). Hence, a j ( k )= 0for 0» j» ffi(`) 1, and the reult follow. Ξ *The firt author wa upported by NSF Grant DMS and NSA Grant MDA Typeet by AMS-TEX 1

2 Corollary 1. Let f(x) 2 Z[x]. Let m be a poitive integer for which Φ m (x) divide f(x). Let f(xy) mod Φ m (y) = Then a j (1) = 0 for 0» j» ffi(m) 1. ffi(m) 1 X a j (x)y j : The Corollary immediately follow by conidering the cae ` = m and k = 1 in Theorem 1. We note that the convere of both Theorem 1 and thi Corollary hold. In other word, if the a j (x) are defined a in Theorem 1 and if Φ k (x) divide each a j (x), then Φ m (x) divide f(x) (provided k and ` are relatively prime); thi follow along line imilar to the argument given for Theorem 1. Alo, we note that the polynomial a j (x) defined in Theorem 1 will necearily have integer coefficient ince Φ`(y) i monic. In addition, we oberve that if ` = 1 in Theorem 1, then the um on the right ide of (1) conit of one term, namely a 0 (x) =f(x). Theorem 2. Let f(x) 2 Z[x] have N non-zero term. Suppoe n i a poitive integer uch that Φ n (x)jf(x). Suppoe further that p 1 ;p 2 ;:::;p k are ditinct prime atifying 2+ kx j=1 (p j 2) >N: Let e j be the non-negative integer for which p e j j jjn. Then for at leat one j 2f1;2;:::;kg, we have Φ m (x)jf(x) where m = n=p e j j. Proof. We firt decribe and P then make ue of Theorem 5 from [1]. For r a poitive integer, define fl(r) =2+ pjr (p 2). Following [1], we call a vanihing um S minimal if no proper ubum of S vanihe. We will be intereted in um S = P t j=1 a j! j where t i a poitive integer, each a j i a non-zero rational number and each! j i a root of unity. We refer to the reduced exponent of uch an S a the leat poitive integer r for which (! i =! 1 ) r =1for all i 2f1;2;:::;tg. Theorem 5 of [1] aert then that if S = P t j=1 a j! j i a minimal vanihing um, then t fl(r) where r i the reduced exponent of S. (Alo, note that Theorem 1 of [1] implie that the reduced exponent r of a minimal vanihing um i necearily quarefree.) To prove Theorem P 2, we uppoe a we may that e j > 0 for each j 2f1;2;:::;kg. We write f(x) = i=1 f i(x) where each f i (x) ia non-zero polynomial diviible by Φ n (x), no two f i (x) have term involving x to the ame power, and i maximal. Thu, each f i ( n ) i a minimal vanihing um. For each i 2f1;2;:::;g,we write f i (x) =x b i g i(x d i ) where b i and d i are non-negative integer choen o that g i (0) 6= 0 and the greatet common divior of the exponent appearing in g i (x) i 1. Then g i ( d i n ) i a minimal vanihing um with reduced exponent n= gcd(n; d i ). If t i denote the number of non-zero term of g i (x), we deduce from Theorem 5 of [1] that N = X i=1 t i X i=1 fl n gcd(n; d i ) 2 + kx j=1 (p j 2) fi fiρ fi fi 1» i» : p j divide fffi n fififi : gcd(n; d i ) 2

3 The inequality in Theorem 2 implie that at leat one of the expreion jf1» i» : p j j(n= gcd(n; d i ))gj i zero. In other word, for ome j 2 f1; 2;:::;kg and every i 2 f1; 2;:::;g,wehavep e j j jd i. Setting m = n=p e j j and d 0 i = d i=p e j j,we obtain that g i( d0 i m )=0. Since gcd(m; p j )=1, pe j j m i a primitive mth root of unity and we deduce g i ( d i m )=0. A thi i true for every i 2f1;2;:::;g,we conclude f( m ) = 0, etablihing the theorem. Ξ Corollary 2. Let f(x) 2 Z[x] have N non-zero term. If f(x) i P diviible by a cyclotomic polynomial, then there i a poitive integer m uch that 2+ pjm (p 2)» N and Φ m (x)jf(x). The above i a direct conequence of Theorem 2. Oberve that it follow eaily from Corollary 2 that if f(x) i diviible by a cyclotomic polynomial, then there i a poitive integer m uch that every prime divior of m i» N and Φ m (x)jf(x). 2. The Algorithm We now decribe how to determine if a given f(x) 2 Z[x] ha a cyclotomic divior. The algorithm we decribe ha been implemented uing MAPLE V, Releae 5, and an interactive World Wide Web page that allow uer to input polynomial f(x) and make ue of the algorithm i available at the web addre The web page wa developed jointly by Dougla Meade and the firt author. A copy of the MAPLE program can alo be downloaded there (but acce to MAPLE i neceary to take advantage of the downloaded program). In particular, we note that on a Sun Ultra 1, the program handle a random polynomial with 100 non-zero term, with coefficient bounded by and with degree in approximately two and a half minute. It i alo poible to run the program with polynomial of degree up to Corollary 2 implie that we need only conider the poibility that Φ m (x)jf(x) where each prime divior of m i» N, the number of non-zero term of f(x). For each prime p» N, we compute the value of» log deg f r(p) = +1: log p Oberve that if p e jm, then deg f(x) deg Φ m (x) =ffi(m) ffi(p e )=p e 1 (p 1) p e 1 o that e» r(p). When computing the a j (x) aociated with Theorem 1 or determining whether f(x) i diviible by a cyclotomic polynomial Φ`(x) in thi ection, we reduce the exponent of f(x) modulo ` (a in [2]) and then perform a diviion by Φ`(x). To bemore precie, uppoe (2) f(x) = rx b j x d j 3

4 where the b j denote non-zero integer and the d j denote ditinct non-negative integer. To compute f(xy) modφ`(y), one can firt compute the value of d j = d j mod ` to obtain (3) f(xy) rx b j x d j yd j X` 1 i=0 c i (x)y i (mod Φ`(y)); where the c i (x) are obtained by combining equal value of d j. Note that thi definition of the c i (x) implie that f(x) = P` 1 c i(x). Now, each of y d j mod Φ`(y) can be computed directly. We will be intereted in the cae that ` = p e for ome prime p and ome poitive integer e. If d j < ffi(`) = p e 1 (p 1), then the value of y d j will remain unchanged when we conider it mod Φ`(y). If d j p e 1 (p 1), then we ue the reduction X p 1 (4) y d j mod Φ`(y) = u=1 y d j p e 1 u ; P which follow from Φ`(y) = Φ p (y pe 1 p 1 ) = ype 1j. Oberve that ince d j < p e, the exponent appearing on the right of (4) are in the interval [0;p e 1 (p 1)). Alo, the exponent on the right of (4) are different for different choice of d j in [p e 1 (p 1);p e ) (a the integer in thi interval are incongruent modulo p e 1 ). A imple algorithm for teting f(x) for diviibility by a cyclotomic polynomial can be decribed a follow. Let p 1 ;p 2 ;:::;p r denote the complete lit of prime» N with p 1 <p 2 < <p r. Conider each integer (5) m = ry j=1 p e j j where 0» e j» r(p j ); and check directly if Φ m (x)jf(x). If ome Φ m (x) divide f(x), then the algorithm top and indicate o. Otherwie, the algorithm output that f(x) ha no cyclotomic divior. Thi imple algorithm i not, however, a efficient awewould like. In particular, to handle polynomial a decribed in the firt paragraph of thi ection would require teting over different value of m. Some improvement can be made by conidering only m for which ffi(m)» deg f(x), but our main improvement will be of a different nature. Given a poitive integer `, conider a j (x) a defined in (1). Oberve that if ome a j (x) ha only one non-zero term, then it cannot be diviible by Φ k (x) for any poitive integer k. It follow then from Theorem 1 that Φ k`(x) doe not divide f(x) for any poitive integer k. We ue thi information to reduce the number of m we need conider in (5). We now decribe how thi i achieved. For 1» j» r, define B j = r(p j ). Let S = f(e 1 ;e 2 ;:::;e r ):0»e j»b j for 1» j» rg: Thu, S conit of the r-tuple formed from the exponent appearing in (5). Define a lexicographical ordering on the element of S. More preciely, if A = (a 1 ;:::;a r ) and 4

5 B =(b 1 ;:::;b r ) are element of S, then we aya<bif there i ome i 2f0;1;:::;r 1g uch that a 1 = b 1 ;a 2 = b 2 ;:::;a i = b i, and a i+1 <b i+1. If d and m are poitive integer, we ay that d exactly divide m and write djjm if djm and gcd(d; m=d) = 1. Finally, we ay that d i obtained from an element (e 1 ;e 2 ;:::;e r ) of S if d = p e 1 1 pe r We decribe a ubroutine hrinkf that make ue of Theorem 1. The input for hrinkf conit of a polynomial g(x), a prime p, and a poitive integer r. We conider g(x) in place of f(x) and ` = p r in Theorem 1. The ubroutine hrinkf compute the a j (x) given there and output a non-zero value of a j (x) which ha a few non-zero term a poible. The idea i that if we wih to know if there i an m exactly diviible by p r for which Φ m (x) divide g(x), then hrinkf(g; p; r) will output a polynomial a(x) which typically" ha le (and never ha more) non-zero term than g(x) and which i diviible by Φ m=p r(x). Suppoe d i obtained from (e 1 ;e 2 ;:::;e r ) in S. Write d = q e0 1 1 qe0 where each q j i prime with q 1 < <q»nand where each e 0 j i a poitiveinteger. Setting f 0(x) =f(x), we define recurively the polynomial f j (x) =hrinkf(f j 1 ;q j ;e 0 j ) for 1» j» : If Φ m (x)jf(x) and djjm, then Theorem 1 implie inductively that Φ (x) divide m ffi q e0 1 1 qe0 j j f j (x). In particular, in thi cae, we have Φ m=d (x)jf (x). Thu, if f (x) i a polynomial with exactly one non-zero term and Φ m (x)jf(x), then it follow that d cannot exactly divide m. We alo oberve that if Φ d (x)jf(x), then f (1) = 0. We wih to determine if f(x) i diviible by Φ m (x) for ome integer m obtained from an element of S. We go through the element of S in increaing order a follow. We begin with the mallet element (0; 0;:::; 0). We let d denote the integer obtained from the element ofsunder conideration; initially then d =1. For each element (e 1 ;:::;e r )of Sunder conideration and the correponding d obtained from it, we proceed by computing f (x) a defined above (in the cae of d = 1, we interpret f (x) to be f 0 (x) = f(x)). If f (x) i a polynomial with exactly one non-zero term, then we know that f(x) cannot be diviible by Φ m (x) for any integer m exactly diviible by d. Thi i where a aving over the imple algorithm decribed earlier in thi ection occur; we can now ignore integer m obtained from element of S if djjm. To explain how thi i done, we let i and j be integer with 0» i» j» r atifying 0» e i <B i ; e i+1 = B i+1 ;e i+2 = B i+2 ;:::;e j = B j ; e j+1 =0;e j+2 =0;:::;e r =0: Here, if a ubcript i not in the range [1;r], then it i to be ignored; and if j = i, then we take thi to mean that the middle equation e t = B t with i+1» t» j are not preent. The cae d = 1 we equate with i = j =0. If i 1, then the next element of S we conider i (e 1 ;:::;e i 1 ;e i +1;0;0;:::;0). Given the ordering of the element of S decribed earlier, if m i obtained from an element ofsbetween (e 1 ;:::;e r ) and (e 1 ;:::;e i 1 ;e i +1;0;:::;0), then djjm. Thi implie that we have kipped over conidering element of S which we know correpond to cyclotomic polynomial that cannot divide f(x). If i = 0 above, then we top the algorithm and indicate that f(x) ha no cyclotomic divior. In thi cae, the remaining element of S correpond to integer m for which djjm and o they need not be conidered. Thi decribe how we proceed if f (x) ha exactly one non-zero term. r. 5

6 Suppoe now that (e 1 ;:::;e r ) and d are a before but f (x) ha more than one non-zero term. We check if f (1) = 0. If it doe, then it might happen that Φ d (x)jf(x) and we determine whether thi i the cae by a direct computation. If we determine Φ d (x)jf(x), then we top and output that f(x) i diviible by a cyclotomic polynomial. Otherwie, either we have f (1) = 0 and Φ d (x) - f(x) orwehave f (1) 6= 0(and Φ d (x) - f(x) here a well). Note that we could have imply checked if Φ d (x)jf(x) without checking the value of f (1); the idea behind checking if f (1) = 0, however, i that we expect uually f (1) 6= 0 and checking the value of f (1) ave time (i.e., diviion by Φ d (x) i more time conuming). If Φ d (x) - f(x), then we imply conider the next element of S greater than (e 1 ;:::;e r ) in our ordering of the element of S. (Some aving i gained by oberving that the f (x) aociated with each ubequent d can be computed by a ingle call to hrinkf by making ue of previouly computed f j (x).) The above decribe the algorithm. To jutify the algorithm work i relatively imple. Firt, oberve that if the algorithm indicate f(x) ha a cyclotomic divior, it doe a thi indication will be a a reult of a direct check of a poible cyclotomic divior a in the paragraph above. On the other hand, for each m a in (5), either the algorithm check if Φ m (x)jf(x) when d = m (directly or by computing f (1)) or the algorithm ha kipped over conidering Φ m (x) a a divior of f(x) becaue the algorithm ha verified that Φ m (x) - f(x) through the ue of ome exact divior d of m. Thu, the algorithm determine if there i an m a in (5) with Φ m (x) dividing f(x), and thi i ufficient for deciding whether f(x) i diviible by a cyclotomic polynomial. 3. Running Time and Other Concluding Remark One difficulty in etimating the running time of the algorithm in Section 2 i in finding good bound for the total number of element of S that are kipped over a we determine d for which Φ m (x) doe not divide f(x) for poitive integer m exactly diviible by d. In addition, we checked whether Φ m (x) divide f(x) by a direct diviion. Even after reducing the exponent of f(x) modulo m, the diviion can cot coniderable time a the ize of m can be large and can even exceed deg f(x). To get a good theoretical bound for the maximal running time of an algorithm which determine whether f(x) ha a cyclotomic factor, we make the following change in the algorithm jut decribed. We alter hrinkf o that, given input g(x), p, and r a before, hrinkf return not jut one but every a j (x) obtained from Theorem 1 (dicarding multiplicitie). If there are t ditinct non-zero a j (x), then the output i a t-tuple with component coniting of thee a j (x) (in any order). We go through the element of S uing their lexicographical ordering a before, checking to ee if Φ m (x) divide f(x) for ome m obtained from an element ofs. Let d be a number we are conidering, obtained from (e 1 ;e 2 ;:::;e r )ins. In particular, we may uppoe that at thi point in the algorithm, we have already determined that each d 0 obtained from a previou element of S i uch that Φ d 0(x) doe not divide f(x) (otherwie the algorithm would have already terminated indicating that f(x) ha a cyclotomic divior). We write d = q e0 1 1 qe0 where, a before, each q j i prime with q 1 < <q»n and each e 0 j i a 6

7 poitive integer. We check initially whether (6) 2+ X j=1 (q j 2)» N: If not, then Corollary 2 implie that we need not conider any m obtained from an element of S that i diviible by d. A in the previou ection, we can kip over thoe element of S exactly diviible by d (and, in fact, more). We conider next the cae where (6) hold. Define a poitive integer t 1 and polynomial f (1) 1 (x);:::;f(t 1) 1 (x) by (1) f 1 (x);:::;f(t 1) 1 (x) = hrinkf(f; q 1 ;e 0 1 ): Next, we compute hrinkf(f 1 ;q 2;e 0 2 ) for each i 2 f1;:::;t 1g and concatenate the reult to obtain f (1) 2 (x);:::;f(t 2) 2 (x) o that each component of each hrinkf(f 1 ;q 2;e 0 2 ) occur a ome f 2 (x) (multiplicitie dicarded). We continue in thi manner until we obtain f (1) (x);:::;f (t ) (x), the value of hrinkf(f 1 ;q ;e 0 ) concatenated. The revied algorithm now read the ame a before except, in the cae that f (1) = 0, we do not check if Φ d (x)jf(x) by a direct computation. Intead we check if every f (1) = 0. We claim that Φ d (x)jf(x) ifand only if f (1) = 0 for every i 2f1;2;:::;t g. The definition of hrinkf and the convere of Corollary 1 imply that if every f (1) = 0, then each f (x) i diviible by Φ 1 (x). Now, the convere of Theorem 1 implie that each q e0 f (x) 2 i diviible by Φ (x), and continued application of the convere of Theorem 1 imply q e0 1 1 qe0 Φ d (x)jf(x). The argument to ee that Φ d (x)jf(x) implie every f (1) = 0 i imilar but in revere. Thu, if (6) hold, we imply perform the algorithm a decribed in Section 2 replacing each check to ee if Φ d (x)jf(x) by checking intead if f (1) = 0 for every i 2 f1; 2;:::;t g. (A noted there ome aving i gained by uing information obtained from previou call to hrinkf; in particular, for any given d a above, the value of f j (x) with j <will be known prior to conidering d.) An etimate for the running time of the algorithm a revied here can be een to depend heavily on the number P of different f j (x) that need to be computed. If the input polynomial i f(x) = N j=1 a jx d j where the a j are nonnegative integer and the d j are integer atifying 0» d 1 <d 2 < <d N, then a determinitic upper bound for the running time i fi exp (2 + o(1)) p N p log N + log log dn log max 1»j»N fja jj +1g a N tend to infinity. Thi etimate i hampered mainly by a preumably poor etimate for the number of element of S kipped over in the approach decribed in Section 2. In any cae, with N fixed, one ee that the running time depend only logarithmically on the degree of f(x). 7

8 We note that H. W. Lentra [2, Propoition 3.5] ha decribed an algorithm which take a given lacunary polynomial and determine all cyclotomic factor, with their multiplicitie, having degree le than ome precribed amount. It would be of interet to find an efficient algorithm for determining all the cyclotomic divior of a lacunary polynomial f(x). We would alo be intereted in good upper bound for the number of m conidered in the algorithm decribed here (excluding the m we kipped over by conideration of an exact divior d of m). Reference 1. J. H. Conway and A. J. Jone, Trignometric diophantine equation (On vanihing um of root of unity), Acta Arith. 30 (1976), H. W. Lentra, Finding mall degree factor of lacunary polynomial, preprint. 3. H. B. Mann, On linear relation between root of unity, Mathematika 12 (1965),