THE AVERAGE NORM OF POLYNOMIALS OF FIXED HEIGHT

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1 THE AVERAGE NORM OF POLYNOMIALS OF FIXED HEIGHT PETER BORWEIN AND KWOK-KWONG STEPHEN CHOI Abstract. Let n be any integer and ( n ) X F n : a i z i : a i, ± i be the set of all polynoials of height and degree n. Let β n() : X 3 n+ P. P F n Here P is the th power of the L nor on the boundary of the unit disc. So β n() is the average of the th power of the L nor over F n. In this paper we give exact forulae for β n() for various values of. We also give a variety of related results for different classes of polynoials including: polynoials of fixed height H, polynoials with coefficients ± and reciprocal polynoials. The results are surprisingly precise. Typical of the results we get is the following. Theore.. For n, we have β n(2) 2 (n + ), 3 and β n(4) 8 9 n n β n(6) 6 9 n3 + 4n n Introduction We explore a variety of questions concerning the average nor of certain classes of polynoials on the unit disk. The Littlewood polynoials L n are defined as follows: { n L n : a i z i : a i ±. Now let i µ n () : 2 n+ P Date: Deceber, Matheatics Subject Classification. Priary C8, 26C5. Key words and phrases. Polynoials of height, Littlewood polynoials, average L p nor. Research of P. Borwein supported by MITACS and by NSERC of Canada and research of K-K Choi supported by NSERC of Canada. Research of Stephen Choi was supported by NSERC of Canada.

2 2 PETER BORWEIN AND KWOK-KWONG STEPHEN CHOI be the average of the th power of the L nors over L n. Here and throughout { P : P (z) dθ, (z e iθ ) is the L nor of P over the unit circle. There are any liiting results concerning expected nors. For exaple the expected nors of rando Littlewood polynoials P of degree n, satisfy li n and for their derivatives li n n /2 n /(2r+)2 { 2 n+ { P 2 n+ P (r) Γ( + /2) Γ( + /2) (2r + ) /2. See [4] for a ore detailed and precise discussion of these results. In this paper, we are interested in finding exact forulae for µ n (). This is the content of the first section where results like the following are obtained. Theore.. For n, we have and µ n (2) n +, µ n (4) 2n 2 + 3n +, µ n (6) 6n 3 + 9n 2 + 4n + µ n (8) 24n 4 + 3n 3 + 4n 2 + 5n + 4 3( ) n. That µ n (2) n+ is trivial since P 2 n+ for each P L n. The above result for µ n (4) is due to Newan and Byrnes [9] though we offer a different proof. The results for µ n (6) and µ n (8) are new and are the tip of an iceberg that we explore further, but by no eans exhaustively, in this paper by extending the results to: height H polynoials; reciprocal polynoials; polynoials with various roots of unity as coefficients. What is striking, and perhaps surprising, is that such exact forulae exist at all. One interesting generalization is the following. Let { n F n (H) : a i z i : a i H, a i Z be the set of all the polynoials of height H of degree n. Let β n (, H) : (2H + ) n+ P i P F n(h) which is now the average over all polynoials of degree n and height at ost H of the th power of the L nor. For this class we get results like β n (4, H) 2 9 H2 (H + ) 2 n H(H + )(9H2 + 9H 3)n + 5 H(H + )(3H2 + 3H ).

3 THE AVERAGE NORM OF POLYNOMIALS OF FIXED HEIGHT 3 There is a considerable literature on the axiu and iniu nors of polynoials in L n. In the L 4 nor this proble is often called Golay s Merit Factor proble. See [5, 3, ]. The specific old and difficult proble is to find the iniu possible L 4 nor of a polynoial in L n. The cognate proble in the supreu nor is due to Littlewood [7, 6]. Both of these probles are at least 5 years old and neither is solved though there has been significant progress on both. For further inforation on these and related probles see [2]. Exact forulae for µ n () for larger values for are obtained in [8]. 2. Average of L 2, L 4 and L 6 As in the previous section let n be any integer and { n L n : a i z i : a i ± i be the set of all Littlewood polynoials of degree n. Let µ n () : 2 n+ P be the average of the th power of the L nor over L n. We are interested in finding exact forulae for µ n (). For any coplex nuber z on the unit circle and any real nuber h, we have (2.) z + h 2 + z h 2 2( z 2 + h 2 ); z + h 4 + z h 4 2( z 4 + 4h 2 z 2 + h 4 + h 2 (z 2 + z 2 )); z + h 6 + z h 6 2( z z 4 h z 2 h 4 + h 6 ) +6( z 2 + h 2 )h 2 (z 2 + z 2 ). Hence for any polynoial P (z), (2.2) zp (z) + h zp (z) h 2 2 2( P (z) h 2 ); zp (z) + h zp (z) h 4 4 2( P (z) h 2 P (z) h 4 ); zp (z) + h zp (z) h 6 6 2( P (z) P (z) 4 4h 2 +9 P (z) 2 2h 4 + h 6 ) + 6h2 P (z) 2 (z 2 P (z) 2 + z 2 P (z) 2 )dθ. Here we have used that z 2 P (z) 2 dθ. Since every polynoial in L n can be written in the for of zp (z) + h with h ± and P L n, we have (2.3) P ( zp (z) + + zp (z) ). Theore 2.. For n, we have (2.4) µ n (2) n +, (2.5) µ n (4) 2n 2 + 3n + and (2.6) µ n (6) 6n 3 + 9n 2 + 4n +.

4 4 PETER BORWEIN AND KWOK-KWONG STEPHEN CHOI Proof. Forula (2.4) is trivial because every Littlewood polynoial of degree n has constant L 2 nor n +. However we give a siple inductive proof because it is indicative of the basic ethod behind all the proofs. Forula (2.5) was first proved by Byrnes and Newan in [9] but we give a sipler proof here. Using (2.3) with 2 and (2.2), we have ( µ n (2) zp (z) n+ 2 + zp (z) 2 2) (2.7) 2 n+ µ n (2) + 2( P ) for any n. It is clear that µ () for any. Thus fro (2.7) we have µ n (2) µ (2) + n n +. This proves (2.4). Siilarly, using (2.2) and (2.3) with 4, we have µ n (4) ( 2 n+ 2 P P ) µ n (4) + 4µ n (2) + for any n. Using (2.4) and µ (4), we get (2.5). To prove (2.6), we need the following two leas. Lea 2.2. For, we have P (z) 2 z dθ. Proof. We prove this result by induction on n. When n, we have P L P (z) 2 z dθ 2 z dθ because z dθ when. Now, by writing every polynoial in L n as zp (z) ± for soe P L n and using (2.) and the induction assuption, we have P (z) 2 z { dθ zp (z) zp (z) 2 z dθ P L n 2 { 2 P (z) z dθ P (z) 2 z dθ.

5 THE AVERAGE NORM OF POLYNOMIALS OF FIXED HEIGHT 5 Lea 2.3. For, we have P L n P (z) 2 z P (z) 2 dθ. Proof. We again prove the lea by induction on n. Like Lea 3.2, the case n is easy. Now P (z) 2 z P (z) 2 dθ { zp (z) + 2 z (zp (z) + ) 2 + zp (z) 2 z (zp (z) ) 2 dθ. We expand the integrand out and after soe siple cancellation we get that the above expression is 2 { 2 P (z) 2 z +2 P (z) P (z) 2 z + z (6z 2 P (z) 2 + 2) dθ P (z) 2 z +2 P (z) 2 dθ + 6 P (z) 2 z dθ because z (6z 2 P (z) 2 + 2) is a polynoial with zero constant ter in z. Now the above integrals are equal to by Lea 2.2 and the induction assuption. We now prove (2.6). Fro (2.2), (2.4), (2.5) and Lea 2.3 { µ n (6) zp (z) n+ 6 + zp (z) 6 6 { 2 n+ 2 P P P µ n (6) + 9µ n (4) + 9µ n (2) + 6n 3 + 9n 2 + 4n +. This copletes the proof of the theore. Let n be any integer and { n F n : a i z i : a i, ± be the set of all the polynoials of height of degree n. Let β n () : 3 n+ P. P F n i As in Theore 2., one can also obtain exact forulae for β n (). The additional details involve observing that since zp (z) + P (z)

6 6 PETER BORWEIN AND KWOK-KWONG STEPHEN CHOI equations (2.2) can be extended easily to allow suing over all the height one polynoials. For exaple, for any polynoial P (z), the iddle identity of (2.2) becoes zp (z) + h zp (z) h zp (z) P (z) h 2 P (z) h 4. Theore 2.4. For n, we have β n (2) 2 (n + ), 3 and β n (4) 8 9 n n β n (6) 6 9 n3 + 4n n It is worth noting that the above technique can also be used to copute the averages of the nors of polynoials of height H. For exaple, one can show the following. Let n and H be integers and let { n F n (H) : a i z i : a i H, a i Z be the set of all the polynoials of height H and degree n. Let β n (, H) : i (2H + ) n+ Theore 2.5. For n and H, we have P F n(h) β n (2, H) H(H + )(n + ), 3 P. and β n (4, H) 2 9 H2 (H + ) 2 n H(H + )(9H2 + 9H 3)n β n (6, H) 2 9 H3 (H + ) 3 n H2 (H + ) 2 (3H 2 + 3H )n H(H + )(3H2 + 3H ) + 35 H(H + )(64H H H 2 8H + 5)n + 2 H(H + )(3H4 + 6H 3 3H + ).

7 THE AVERAGE NORM OF POLYNOMIALS OF FIXED HEIGHT 7 3. Forula for the Average of L 8 The forula for µ n (8) is soewhat ore coplicated. We proceed as follows. Lea 3.. For, we have 2 n+ if n; z 2 P (z) 2 dθ if > n. Proof. This is clearly true for. Suppose. If n, then If > n, then 2 n+ 2 n+ 2 n 2 n z 2 P (z) 2 dθ 2 n + 2 n +. 2 n+ z 2 { (zp (z) + ) 2 + (zp (z) ) 2 dθ z 2 (z 2 P (z) 2 + )dθ z 2( ) P (z) 2 dθ z 2 P (z) 2 dθ 2 2 P (z) 2 dθ P L P L z 2( n) P (z) 2 dθ z 2( n) dθ. Lea 3.2. For, we have (3.) 2 n+ n + if n; z 2 P (z) 4 dθ if > n.

8 8 PETER BORWEIN AND KWOK-KWONG STEPHEN CHOI Proof. Let B n () be the left hand side of (3.). By Lea 2.2 and (2.) B n () 2 n+ z 2 ( zp (z) zp (z) 2 )dθ P L n where 2 n B n () + 2 n B n () + C n ( ) C n () : 2 n+ P L n by Lea 3.. Clearly, B () z 2 ( P (z) p(z) z 2 P (z) 2 + z 2 P (z) 2 )dθ z 2( ) P (z) 2 dθ if n; z 2 P (z) 2 dθ if > n; z 2 dθ. For n, we have B n () B n ()+C n ( ). If > n, then C n ( ) and hence B n () B (). If n, then B n () B n () + C n ( ) C ( ) + C ( ) + + C n ( ) n +. Lea 3.3. For, we have 2 n+ P (z) 4 z 2 P (z) 2 dθ P L n if > n; 3 2 (n ) ( )+n 2 (n ) + 2 if n. Proof. Let Then A n () 2 n+ A n () : 2 n+ P (z) 4 z 2 P (z) 2 dθ. z 2 { zp (z) + 4 (zp (z) + ) 2 + zp (z) 4 (zp (z) ) 2 dθ. We expand the integrand out and use Leas 3. and 3.2. We derive that for n,, A n () A n ( + ) + 6B n () + C n ( ).

9 THE AVERAGE NORM OF POLYNOMIALS OF FIXED HEIGHT 9 It is clear that A (). Thus if > n, then B n () C n ( ) and hence A n () A ( + n). Suppose n. If n + (od 2), then A n () A n 2 ( + 2) + 6(B n () + B n 2 ( + )) + (C n ( ) + C n 2 ()) ( ) ( ) n + n + A n+ + 6B n () + 6B n 2 ( + ) + + 6B +n ( ) + n +C n ( ) + C n 2 () + + C +n Since A () 6B () + C ( ) by Leas 3. and 3.2, we see that A n () + 6( (n )) + n (n )2 + 7 (n ) +. 2 The case n + (od 2) can be proved in the sae way. In particular, A () and for n (3.2) A n () 3 2 n2 + 2 n + ( )n. 2 We now coe to the proof for the forula of µ(8). Since zp (z) zp (z) 8 2( P (z) P (z) P (z) P (z) 2 + ) + 2(6 P (z) 4 z 2 P 2 (z) + 6 P (z) 4 z 2 P 2 (z) + 6 P (z) 2 z 2 P 2 (z) + 6 P (z) 2 z 2 P 2 (z)) It follows fro Leas 2.3 and 3.3 that + 2(6P 2 (z)z 2 + 6P 2 (z)z 2 + P (z) 4 z 4 + P (z) 4 z 4 ). µ n (8) µ n (8) + 6µ n (6) + 36µ n (4) + 6µ n (2) + + 2A n (). In view of Theore 2. and (3.2), we have µ n (8) µ n (8) + 6(6n 3 9n 2 + 4n) + 36(2n 2 n) + 6n + Thus we have proved +8n 2 3n + 6 6( ) n 24n 4 + 3n 3 + 4n 2 + 5n + 4 3( ) n. Theore 3.4. For n, we have 2 n+ P n 4 + 3n 3 + 4n 2 + 5n + 4 3( ) n. 4. Derivative and reciprocal polynoials If we replace z by z/w in (2.) then we have a hoogeneous for of (2.) z + hw 2 + z hw 2 2( z 2 + h 2 w 2 ); (4.) z + hw 4 + z hw 4 2( z 4 + 4h 2 z 2 w 2 + h 4 w 4 +h 2 w 4 ( ( ) z 2 ( w + z ) 2)). w Let P () (z) be the th order derivative of P (z).

10 PETER BORWEIN AND KWOK-KWONG STEPHEN CHOI Theore 4.. For n, we have if > n; (4.2) 2 n+ P () 2 2 n ( ) 2 l! 2 if n; l and 2 n+ P () 4 4 P L n if > n; ( n ( ) ) 2 2 l n ( ) (4.3) 4 l 2! 4! 4 if n. l Proof. We write every polynoial in L n as P (z) ± z n for soe P L n. In view of (4.), we have 2 n+ 2 n+ 2 n P () n! 2 n l ( l ( ) P () (z) + (z n ) () P () (z) (z n ) () 2 2 P () 2 2 +! 2 ( n P () 2 2 +! 2 ( ) 2 l. ( ) 2 n ) 2 ) This proves (4.2). For (4.3), we have 2 n+ P () 4 4 ( ( ) ( ) ) n n 2 n+ P () (z) +! z n P () (z)! z n 4 4 ( ( ) 2 ( ) ) 4 n n 2 n P () ! 2 P () 2 2 +! 4 because P () (z) is a polynoial of degree less than n for any P L n and hence (P () (z)/z n ) 2 dθ. Using (4.2), we have 2 n+ P () n P () ! 4 ( n ) 2 n l ( ) 2 ( ) 4 l n +! 4.

11 THE AVERAGE NORM OF POLYNOMIALS OF FIXED HEIGHT As before, this proves (4.3). A polynoial P (z) of degree n is reciprocal if P (z) P (z) and negative reciprocal if P (z) P (z) where P (z) z n P ( z ). In view of (4.), we have P (z) + z n+ P (z) P (z) z n+ P (z) P 4 4. Since every reciprocal and negative reciprocal polynoial of degree n 2 + can be written as P (z) + z + P (z) and P (z) z + P (z) for soe P L respectively, so the average of P 4 4 over the reciprocal and negative reciprocal Littlewood polynoials in L n is 2 +2 P L P L P 4 4 { P (z) + z + P (z) P (z) z + P (z) 4 4 6( ) 3n 2 + 3n by Theore.. Now since P Ln P P P 4 4 we have proved that if n is odd P (z) + z + P (z) 4 4 P L P (z) z + P (z) 4 4 P Ln P P P 4 4 P L P Ln 2 (n+)/2 P P P 4 4 P Ln 2 (n+)/2 P P P 4 4 3n 2 + 3n. By a siilar arguent, we can show that the average P 4 4 over the reciprocal Littlewood polynoials in L n is 3n 2 + 3n + if n is even. 5. polynoials with coefficients of odulus The critical lea in this analysis is Lea 5.2 below. Lea 5.. For l, we have (5.) in(l, l) j ( 2j )( 2j j )( ) 2j l j ( ) 2. l

12 2 PETER BORWEIN AND KWOK-KWONG STEPHEN CHOI Proof. Without loss of generality, we ay assue l /2. Thus the left hand side of (5.) becoes l j l j ( 2j )( 2j j )( ) 2j l j! (j!) 2 (l j)!( l j)! ( ) l ( )( ) l l l j j j ( ) 2. l Lea 5.2. Let k and ζ k e i k. Then for any coplex nuber z, we have k k ( ) 2 z (5.2) z + ζ j l 2l l, if k < ; k 2 j k ( ) 2 z l 2l l + (z + z ), if k. Proof. We first have k z + ζ j k 2 j k (z + ζ j k ) (z + ζ j j k ) k ( z (zζ j + zζ j k )) j k j l l l r ( ) l ( l k ( z 2 + ) l )( l r l r ( ) l (zζ j k r )r (zζ j k )l r k )( z 2 + ) l z r z l r j ζ j(l 2r) k. Since k j ζja k equals if a (od k) and equals to k otherwise, if < k, then we have k )( ) z + ζ j k 2 k r l( l l 2r (od k) ( z 2 + ) l z r z l r l r j l k )( ) l ( l l is even ( z 2 + ) l z l l l/2 ( )( ) 2l k ( z 2 + ) 2l z 2l. 2l l l /2

13 THE AVERAGE NORM OF POLYNOMIALS OF FIXED HEIGHT 3 We then expand the ter ( z 2 + ) 2l to get k ( )( z + ζ j 2l k 2 k 2l l j l /2 k j k k j j ( j by Lea 5.. If k, then k z + ζ j k 2 k j This proves our lea. l /2 l /2 l in(j, j) ) 2 z 2j ( 2l ) 2l r ( 2l r r 2l( r+lj )( 2l l ( 2l )( 2l l ) z 2(l+r) )( )( ) 2l 2l 2l l r z 2l )( 2l j l ) z 2l ) ( z 2 + ) 2l z 2l + (z + z ). Let n and { n L n,k : a i z i : a k i i be the set of all polynoials of degree n whose coefficients are kth root of unity. This is a generalization of Littlewood polynoials and clearly L n L n,2. Then we have Theore 5.3. For n and k, we have l l n ( ) 2 ( ) 2 (5.3) k n+ P 2 l 2 l l 2,k l l 2 l n ( ln l n ) 2. Proof. In view of Lea 5.2, the left hand side of (5.3) is because This copletes the proof. k n+ l,k j ( ) 2 l l l l 2 k k zp (z) + ζ j k 2 k n l n l n P 2l 2l,k ( ) 2 ( ) 2 l l l 2 P 2 2. P L,k ( ) 2 ln l n

14 4 PETER BORWEIN AND KWOK-KWONG STEPHEN CHOI For exaple, for k 5, the averages P over the polynoials in L n,k are n +, 2n 2 + 3n +, 6n 3 + 9n 2 + 4n +, 24n n 3 + n 2 + n +, 2n 5 + 5n n 2 44n + when 2, 4, 6, 8, respectively. References [] A.T. Bharucha-Reid and M. Sabandha, Rando Polynoials, Acadeic Press, Orlando, 986. [2] P. Borwein Coputational Excursions in Analysis and Nuber Theory, Springer Verlag 22. [3] P. Borwein and S. Choi, Explicit erit factor forulae for Fekete and Turyn Polynoials. Trans. Aer. Math. Soc. 354 (22), [4] P. Borwein and R. Lockhart, The expected L p nor of rando polynoials, Proc. Aer. Math. Soc. 29 (2), no. 5, [5] M.J. Golay, The erit factor of long low autocorrelation binary sequences. IEEE Transactions on Inforation Theory 28 (982), [6] J-P. Kahane, Sur les polynôes á coefficients uniodulaires. Bull. London Math. Soc. 2 (98), [7] J.E. Littlewood, Soe Probles in Real and Coplex Analysis, Heath Matheatical Monographs, Lexington, Massachusetts, 968. [8] T. Mansour, Average nors of polynoials. Adv. in Appl. Math. 32 (24), no. 4, [9] D. J. Newan and J. S. Byrnes, The L 4 nor of a polynoial with coefficients ±, Aer. Math. Monthly 97 (99), no., [] B. Saffari, Barker sequences and Littlewood s two-sided conjectures on polynoials with ± coefficients, Séinaire d Analyse Haronique. Anneé 989/9, 39 5, Univ. Paris XI, Orsay, 99. Departent of Matheatics, Sion Fraser University, Burnaby, British Colubia, Canada V5A S6 E-ail address: pborwein@@cec.sfu.ca Departent of Matheatics, Sion Fraser University, Burnaby, British Colubia, Canada V5A S6 E-ail address: kkchoi@ath.sfu.ca