On the distribution of coefficients of powers of positive polynomials

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1 AUSTRALASIAN JOURNAL OF COMBINATORICS Volume 49 (2011), Pages O the distributio of coefficiets of powers of positive polyomials László Major Istitute of Mathematics Tampere Uiversity of Techology PL 553, Tampere Filad laszlo.major@tut.fi Abstract Usig the formalism of polyomials with positive coefficiets, the fact that exactly half of all subsets of a fiite set have eve cardiality ca be geeralized asymptotically. The well-kow fact that every fiite oempty set has as may subsets of eve cardiality as of odd cardiality ca be restated as follows. If d deotes the particular iteger 2, the for ay itegers 0 j d 1, 1 ad the stadard expasio c,0 + c,1 x + c,2 x 2 + of the polyomial ( x) we have c,k = 1 d. k j mod d For = 0 this obviously fails. It also fails i geeral if d 3. We will show however, that it remais asymptotically true for all d 2 ad ot oly for the polyomial x, but for ay polyomial with positive coefficiets summig to 1 (Theorem 2 2 1). I particular, this will imply that summig up every d th etry i the th row i Pascal s triagle asymptotically yields 2 /d. Questios of distributio about powers of polyomials with positive coefficiets, i coectio with biomial ad multiomial coefficiets i particular, have bee studied both by classical ad cotemporary authors. Whe Euler (1765) was ivestigatig the properties of the triomial coefficiets (see Adrews [1]), he obtaied a uimodal distributio of the coefficiets by expadig the th power of the polyomial 1 + x + x 2. I the aalogous case of the polyomial 1 + x the correspodig uimodal distributio is the th row of Pascal s triagle. Further results ad refereces cocerig the powers of the polyomial 1 + x ca be foud i Marcus ad Tucel [6]. I geeral the distributio of the coefficiets of the th power of the polyomial p = p 0 + p 1 x + + p m x m with oegative real coefficiets is ot ecessarily

2 240 LÁSZLÓ MAJOR uimodal, but some sufficiet coditios for uimodality of powers of polyomials are give by Odlyzko ad Richmod [7] ad by De Agelis [2]. I this paper we are cocered with a questio of equidistributio of coefficiets. The mai result is Theorem 1. The lemmas ad corollary provide backgroud ad a extesio. Let d 2 be a fixed positive iteger. Let r(p) deote the remaider of the divisio of the polyomial p by x d 1, i.e. that uique polyomial of degree less tha d for which the polyomial p is cogruet to r(p) modulo x d 1. A polyomial with positive (oegative) real coefficiets is said to be positive (oegative) polyomial. The followig Lemma provides a sufficiet coditio for the powers of a oegative polyomial p to have positive remaider by x d 1. Lemma 1 Let p R[x] be a oegative, ozero polyomial. For ay q R[x] ad i N let q i deote the coefficiet of x i i q. If there exist k,l {0,...,d 1} such that r(p) k > 0, r(p) l > 0 ad gcd(d, k l) =1,ther(p ) is a positive polyomial for all iteger >d 1. Proof: It ca be assumed that k<l. Let us deote the differece l k by h. It is easy to see that if r((x k +x l ) d 1 ) is positive the also r(p d 1 ) is positive. I additio, r((x k +x l ) d 1 )=r(x k(d 1) (1+x h ) d 1 ) is positive if r((1+x h ) d 1 ) is positive, so let us cocetrate o the polyomial r((1 + x h ) d 1 ). We expad the expressio (1 + x h ) d 1 i the polyomial rig R[x] by usig the Biomial Theorem: d 1 ( (1 + x h ) d 1 = d 1 ) i (x hi ). (0.1) Let us assume that the followig cogruece holds for some i, j {0,...,d 1}: ( d 1 ) i (x hi ) ( ) d 1 j (x hj ) mod x d 1. (0.2) From the equatio (0.2) it follows that hi hj is divisible by d. But it was assumed that gcd(h, d) = 1, so cosequetly i = j. Therefore r((1 + x h ) d 1 ) has exactly d differet ozero coefficiets, so i other words it is positive ad cosequetly r(p d 1 ) is also positive. It ca be show without difficulty that for ay d 1, the remaider r(p )is positive if r(p d 1 ) is positive. I order to be able to deal more efficietly with powers of polyomials we eed the cocept ad some useful properties of circulat matrices. For a classical referece see e.g. Davis [3]. Let v =(v 0,v 1,...,v d 1 ) be a row vector i R d. The permutatio ρ : R d R d give by ρ(v 0,v 1,...,v d 1 )=(v d 1,v 0,...,v d 2 ) is called a cyclic permutatio. The circulat matrix associated to the vector v is the d d matrix whose i th row is ρ i 1 (v), i =1,...,d ad it is deoted by C = circ(v 0,v 1,...,v d 1 ) = circ(v).

3 DISTRIBUTION OF COEFFICIENTS OF POLYNOMIALS 241 The product of two circulat matrices is circulat, therefore ay positive iteger power of a circulat matrix is circulat. We shall use the coectio betwee powers of circulat matrices ad the powers of residue polyomials. If a =(a 0,a 1,...,a d 1 ) is the coefficiet vector of some polyomial r(p) R[x], the the first row of (circ(a)) is the coefficiet vector of r(p ). Lally ad Fitzpatrick [5] give a geeral overview i this subject. A matrix is called positive (oegative) if all its etries are positive (oegative) real umbers. A d d oegative matrix M is said to be doubly stochastic if the sum of the etries i each row ad i each colum equals 1. The product of doubly stochastic matrices is doubly stochastic. The followig lemma follows from a geeral result o eigevectors of positive matrices, Theorem i Hor ad Johso [4]. Here we provide a direct proof. A matrix whose etries are all oes will be deoted by U. Lemma 2 If M is a d d positive doubly stochastic matrix, the M = 1 U. (0.3) d Proof: Let us deote the doubly stochastic matrix 1 U by J. If M = J, the the d statemet is trivial. Otherwise the matrix M ca be give i the followig form: M = λj +(1 λ)m 0, (0.4) where 0 <λ<1adm 0 is a doubly stochastic matrix. Let us write the th power of M by usig the Biomial Theorem ad the fact that JD = DJ = J for all doubly stochastic matrix D, M = ( i 1 )(λj) i ((1 λ)m 0 ) i =(1 λ) M 0 + J =(1 λ) M 0 + J(1 (1 λ) )=(1 λ) (M 0 J)+J ( ) λ i (1 λ) i i From the above expressio follows our statemet, because 0 < 1 λ<1 ad cosequetly (1 λ) =0. I order that the Lemma 2 could be applied for the powers of polyomials, we shall restrict our attetio to polyomials p whose coefficiets sum to 1 (p(1) = 1). We may do this without loss of geerality, because all polyomials ca be writte i the form p = p(1) p, where p has the metioed property. It ca also be said that the coefficiets of p form a stochastic vector. Clearly the sum of the coefficiets of (p ) is also 1 ad the sum of the coefficiets of p is equal to p(1). By usig Lemma 1, Lemma 2 ad the coectio betwee powers of circulat matrices ad the powers of residue polyomials we obtai the followig theorem:

4 242 LÁSZLÓ MAJOR Theorem 1 Let p be a ocostat polyomial with positive real coefficiets summig to 1 ad d 2 a fixed iteger. The for all 0 j d 1 k j mod d (p ) k = 1 d (0.5) where (p ) 0 +(p ) 1 x +(p ) 2 x 2 + is the stadard expasio of p. The restrictio that p is positive is quite strog. Theorem 1 ca be exteded to larger classes of polyomials. For istace the followig corollary applies Lemma 1 so as to provide such a extesio. Corollary 1 Let d 2 a fixed iteger. If the oegative polyomial p satisfies the coditios of Lemma 1 ad its coefficiets sum to 1, the the covergece (0.5) holds for p, or equivaletly, for all 0 j d 1 r(p ) j = 1 d, where r(p ) j deotes the coefficiet of x j i the remaider of p modulo x d 1. Proof: Let p be a polyomial satisfyig the coditios of Lemma 1 ad let p(1) = 1. Let us deote the coefficiet vector of r(p) byc ad the circulat matrix circ(c) by C. ByLemma1ifm d 1theC m is a positive matrix. We use J to deote the d d matrix i which every etry is 1. By Lemma 2, d C(d 1) = J. We may also express this fact by usig the max orm of matrices: C(d 1) J =0. We show that for all 0 i d 1, C (d 1)+i = J. We recall that JC i = J, so we have C (d 1) C i J = C (d 1) C i JC i C (d 1) J C i. (0.6) Because C i is a costat, we obtai that C (d 1)+i = J for all 0 i d 1, cosequetly C m J as m teds to ifiity. Applicatio to Pascal s triagle For ay d 2 we ca extract from the th row of Pascal s triagle (( ( 0),..., )), d disjoit subsequeces, oe for each j = 0,...,d 1: ( ) ( ) ( ),,,... j j + d j +2d It follows from Theorem 1 that the d differet sums ( ) ( j + ) ( j+d + j+2d) + are asymptotically equal to 2 /d as.

5 DISTRIBUTION OF COEFFICIENTS OF POLYNOMIALS 243 Ackowledgemet I would like to express my sicere gratitude to the referee for the remarks ad for the valuable advice cocerig the refereces. Refereces [1] G.E. Adrews, Euler s Exemplum Memorabile Iductiois Fallacis ad q- Triomial Coefficiets, J. Amer. Math. Soc. 3 (1990), [2] V. De Agelis, Asymptotic expasios ad positivity of coefficiets for large powers of aalytic fuctios, It. J. Math. Math. Sc (16) (2003), [3] P.J. Davis, Circulat Matrices, Joh Wiley ad Sos, [4] R.A. Hor ad C.R. Johso, Matrix Aalysis, Cambridge Uiversity Press, [5] K. Lally ad P. Fitzpatrick, Algebraic Structure of Quasicyclic Codes, Discrete Appl. Math. 111 (2001), [6] B. Marcus ad S. Tucel, Matrices of polyomials, positivity, ad fiite equivalece of Markov chais, J. Amer. Math. Soc. 6 (1993), [7] M. Odlyzko ad L.B. Richmod, O the uimodality of high covolutios of discrete distributios, A. Probab. 13 (1985), (Received 5 July 2010; revised 4 Nov 2010)