Limiting value of higher Mahler measure

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1 Limiting valu of highr Mahlr masur Arunabha Biswas a, Chris Monico a, a Dpartmnt of Mathmatics & Statistics, Txas Tch Univrsity, Lubbock, TX 7949, USA Abstract W considr th k-highr Mahlr masur m k P ) of a Laurnt polynomial P as th intgral of log k P ovr th complx unit circl. In this papr w driv an xplicit formula for th valu of m k P ) /k! as k. Kywords: Mahlr masur, highr Mahlr masur. Introduction For a non-zro Laurnt polynomial P z) C[z, z ], th k-highr Mahlr masur of P is dfind [4] as m k P ) = log k P 2πit) dt. For k = this coincids with th classical log) Mahlr masur dfind as n n mp ) = log a + log max{, r j }), for P z) = a z r j ), j= j= sinc by Jnsn s formula mp ) = m P ) [3]. Though classical Mahlr masur was studid xtnsivly, highr Mahlr masur was introducd and studid vry rcntly by Kurokawa, Lalin and Ochiai [4] and Akatsuka []. It is vry difficult to valuat k-highr Mahlr masur for polynomials xcpt fw spcific xampls shown in [] and [4], but it is rlativly asy to find thir iting valus. In [5] Lalin and Sinha answrd Lhmr s qustion [3] for highr Mahlr masur by finding non-trivial lowr bounds for m k on Z[z] for k 2. addrss: arunabha.biswas@ttu.du Arunabha Biswas) Prprint submittd to Journal of Numbr Thory March 3, 24

2 In [2] it has bn shown using Akatsuka s zta function of [4] that for a =, m k z + a) /k! /π as k. In this papr w gnraliz this rsult by computing th sam it for an arbitrary Laurnt polynomial P z) C[z, z ] using a diffrnt tchniqu. Thorm.. Lt P z) C [ z, z ] b a Laurnt polynomial, possibly with rpatd roots. Lt z,..., z n b th distinct roots of P. Thn m k P ) = k k! π P z j ), z j S whr S is th complx unit circl z =, and th right-hand sid is takn as if P z j ) = for som z j S, i.., if P has a rpatd root on S. 2. Proof of th thorm W first prov svral lmmas which ssntially show that th intgrand may b linarly approximatd nar th roots of P on S. Lmma 2.. Lt P z) C [ z, z ] b a Laurnt polynomial and A [, ] b a closd st such that P 2πit) for all t A. Thn log k P 2πit ) dt = k k! A Proof. Sinc A is closd, du to th priodicity of 2πit and continuity of P 2πit ) thr xist constants b and B such that < b P 2πit ) B on A. Thn for ach positiv intgr k, log k P 2πit ) )/k! is boundd btwn log k b)/k! and log k B)/k!, and thrfor /k!) A logk P 2πit) dt is boundd btwn µa log k b)/k! and µa log k B)/k!, whr µa is th Lbsgu masur of A. Th rsult follows by ltting k tnd to infinity. Lmma 2.2. Lt P z) C [ z, z ] b a Laurnt polynomial with a root of ordr on at z = 2πit, and P z) b its drivativ with rspct to z. Thn for ach ε, ) thr xists δ > such that t t < δ implis 2π ε)t t )P 2πit) P 2πit) 2π + ε)t t )P 2πit). 2

3 Proof. St ft) = P 2πit). Thn f t ) = 2πiP 2πit ) and f t ) = t t ft) ft ) t t. Sinc f t ), it follows that for ach ε, ) thr xists δ > such that < t t < δ implis ε < ft) ft ) t t ) which provs th lmma sinc ft ) = P z ) =. f t ) < + ε, Lmma 2.3. Lt c, and t R. Thn for all ε >, t+ε k k! log k ct t ) dt = 2 c. t ε Proof. For k and x >, it follows from intgration by parts and induction that x log k u du = x log k x + x k j= ) j k! log k j x. k j)! Using th vn symmtry of th intgrand and substituting u = ct t ), w hav k! t+ε t ε log k ct t ) dt = 2 c k! cε log k u du, and it follows that t+ε k k! log k ct t ) dt 2 cε = log k u du t k ε c k! = 2ε log k cε k ) j log k j cε k + k! k j)! j= ) n log n cε = 2ε n! n= = 2ε log cε = 2/ c. Lmma 2.4. Lt P z) C [ z, z ] b a Laurnt polynomial with a root of ordr on at z = 2πit. Thn for all sufficintly small δ >, t+δ log k P 2πit ) dt = k k! π P 2πit ). 3

4 Proof. First notic that sinc z has ordr on, it cannot b a root of P z). Now lt ε, ). By Lmma 2.2 thr is a δ > such that t t < δ implis 2π ε)t t )P 2πit) P 2πit ) 2π + ε)t t )P 2πit). Stting c = 2π ε)p 2πit ) and d = 2π + ε)p 2πit ) it follows that for < t t < δ, and hnc log ct t ) log P 2πit ) log dt t ), log k ct t ) log k P 2πit ) log k dt t ), for all k N. Thrfor, t +δ log k ct t ) dt t +δ log k P 2πit ) t+δ dt log k dt t ) dt. But on t δ, t + δ), for ach fixd k, ithr all thr functions log k ct t ), log k P 2πit ) and log k dt t ) ar ngativ if k is odd), or positiv if k is vn). So th intgrals of thir absolut valus ar qual to th absolut valus of thir intgrals and thrfor w hav t +δ t +δ log k ct t ) dt log k P 2πit ) dt By Lmma 2.3 it follows that 2 c k k! t+δ t +δ log k P 2πit) 2 d. log k dt t ) dt. Sinc c = 2π ε)p 2πit ) and d = 2π+ε)P 2πit ) and ε > is arbitrary, w ar don. With ths lmmas, w now procd to prov th main thorm. Proof of Thorm.. First notic that m k P ) k! = k! log k P 2πit) dt. 4

5 If P z) dos not hav any roots on S thn choosing A = [, ] and applying Lmma 2. w s that m k P ) /k! as k and th thorm holds in this cas. Now lt t,..., t m [, ] such that 2πit,..., 2πitm of P on S. ar th distinct roots Lt δ > b sufficintly small so that P 2πitj ) < on ach intrval t j δ, t j + δ), j =,..., m, and ths intrvals ar disjoint and dfin A = [, ] m t j δ, t j + δ). j= Using Lmma 2., and th fact that log P 2πit ) < on [, ] \ A, w find that m k P ) = log k P 2πit ) dt + log k P 2πit ) dt k k! k k! A [,]\A m tj+δ = log k P 2πit ) dt 2.5) k k! j= t j δ If P has no rpatd roots on S, thn by Lmma 2.4, this final sum is qual π m j= P 2πitj ), and so th thorm is provn in this cas. Finally, if P has a rpatd root on S, w may assum without loss of gnrality that P z ) = P z ) = whr z = 2πit. With ft) = P 2πit ), w hav that ft ) = f t ) =. Thn for ach ε, ) thr is a δ ε, ) such that ft) t t = ft) ft ) t t ε, for all < t t < δ ε. It follows that log ft) log εt t ) < for all < t t < δ ε, and so log k ft) log k εt t ), for all < t t < δ ε. W may assum that δ ε < δ, and using 2.5) and Lmma 2.3 dduc that m k P ) t+δ log k P 2πit ) dt k k! k t δ t+δ = log k P 2πit ) dt k t δ t+δ ε log k εt t ) dt k = 2 ε. 5 t δ ε

6 Sinc ε, ) was arbitrary, th it in qustion divrgs to and th thorm is provn. [] H. Akatsuka, Zta Mahlr masurs, J. Numbr Thory 29 ) 29) [2] A. Biswas, Asymptotic natur of highr Mahlr masur, 23) Submittd). [3] G. Evrst, T. Ward, Hights of polynomials and ntropy in algbraic dynamics, Springr 29). [4] N. Kurokawa, M. Lalín, H. Ochiai, Highr Mahlr masurs and zta functions, Acta Arith. 35 3) 28) [5] M. Lalín, K. Sinha, Highr Mahlr masur for cyclotomic polynomials and Lhmr s qustion, Ramanujan J. 26 2) 2)

cycle that does not cross any edges (including its own), then it has at least

cycle that does not cross any edges (including its own), then it has at least W prov th following thorm: Thorm If a K n is drawn in th plan in such a way that it has a hamiltonian cycl that dos not cross any dgs (including its own, thn it has at last n ( 4 48 π + O(n crossings Th