TWO INEQUALITIES ON THE AREAL MAHLER MEASURE

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1 Illiois Joural of Mathematics Volume 56, Number 3, Fall 0, Pages S TWO INEQUALITIES ON THE AREAL MAHLER MEASURE KWOK-KWONG STEPHEN CHOI AND CHARLES L. SAMUELS Abstract. Recet wor of Pritser defies ad studies a areal versio of the Mahler measure. We further explore this fuctio with a particular focus o the case where its value is small, as this is most relevat to Lehmer s cojecture. I this situatio, we provide improvemets to two iequalities established i Pritser s origial paper.. Itroductio For a polyomial P z with complex coefficiets, the logarithmic Mahler measure of P is give by log MP = π log P e it dt. π 0 I view of the well-ow idetity π. log MP = lim P e it /p dt p, p 0 + π the Mahler measure is ofte called the H 0 Hardy space orm. If P z = N a N = z z, the we may apply Jese s formula to fid that. log MP =log a N + log z, 0 z > where the sum is tae over all roots of P that lie outside the closed uit dis. If we cosider oly polyomials P Z[z], the it follows form. that MP. Kroecer s theorem implies that MP = if ad oly if P is a Received August 6, 0; received i fial form October 4, 0. The research of both authors was supported i part by NSERC of Caada. The secod author thas Simo Fraser Uiversity ad the Uiversity of British Columbia where the majority of this research too place. 00 Mathematics Subject Classificatio. C08, G c 03 Uiversity of Illiois

2 86 K.-K. S. CHOI AND C. L. SAMUELS product of cyclotomic polyomials ad ±z. As part of a famous 933 article costructig large prime umbers, Lehmer [8] ased whether there exists a costat c> such that MP c i all other cases. I his ivestigatios, he oted that lz=z 0 + z 9 z 7 z 6 z 5 z 4 z 3 + z + has Mahler measure equal to.7..., although he foud o polyomial of measure smaller tha this. I view of this observatio, we are led to the followig cojecture. Cojecture. Lehmer s cojecture. There exists c> such that MP c wheever P Z[z] is ot a product of cyclotomic polyomials ad ±z. Although Lehmer s cojecture remais ope, umerous special cases of it have bee resolved see, for istace, [], [], [], [4]. The best ow uiversal lower boud is 3 log log deg P log MP, log deg P which arises from the wor of Dobrowolsi [3]. Ispired by Lehmer s origial paper, various authors have costructed aalogues ad geeralizatios of the Mahler measure. Dubicas ad Smyth [4], [5] defied ad examied the metric Mahler measure, which is a fuctio m : Q /TorQ [0, that is closely related to the classical Mahler measure. They ote that α, β m αβ defies a metric o Q /TorQ which iduces the discrete topology if ad oly if Lehmer s cojecture holds. A ultrametric versio of m was explored by Fili ad the secod author [6], while both fuctios were later geeralized by the secod author [0], []. I a differet directio, Siclair [3] defied the otio of a multiplicative distace fuctio. This is a map φ : C[z] [, of which the ologarithmic Mahler measure M is a example. Other examples of such fuctios are discussed i [3]. Returig ow to the situatio where P z is a arbitrary polyomial over C, Pritser[9] recetly defied the areal Mahler measure of P C[z] by.3 log P 0 = log P z da, π where D is the closed uit dis. It is oted that P 0 satisfies a aalogue of. log P 0 = lim P z /p da p. p 0 + π D Although we will focus o.3 almost exclusively i this article, we ote the wor of Huag [7] which studies a Foc space versio of the Mahler measure aalogous to P 0. D

3 THE AREAL MAHLER MEASURE 87 I Theorem. of [9], Pritser establishes the importat idetity.4 log P 0 =logmp + z, which we should view as the aalogue of. for the areal Mahler measure. Ideed, by addig ad subtractig z to the right-had side of., we quicly obtai that.5 log MP =log a 0 log z, where a 0 is the costat term of P. Combiig this with.4, we fid that.6 log P 0 =log a 0 + z log z. We iterpret this idetity as the areal Mahler measure versio of.5. I view of.4, we obtai the iequalities see Corollary. i [9].7 deg P +logmp log P 0 log MP. Equality occurs i the first iequality precisely whe P has o zeros except at the origi, ad we obtai equality i the secod iequality wheever P has o zeros iside the ope uit dis. If we wish to study Lehmer s cojecture, the we will most ofte cosider polyomials of small Mahler measure. I this case, the left-had side of.7 is egative, ad therefore, weaer tha eve the most trivial boud that P 0 a 0. Usig oly elemetary techiques, we aticipate that further improvemets to.7 are possible whe MP is small ad P Z[z]. Ideed, examiig the idividual terms of the sums i.5 ad.6, we fid the power series expasios log x = x = Of course, this implies that ad log x = x + x =3 log x = x + x, l l x. so that the expasios for log x ad x / log x have the same mai terms. I view of these observatios, we fid it reasoable to guess that log MP behaves similarly to log P 0 whe these values are close to zero. Nevertheless, these remars are oly heuristic ad must be made precise. I this article, we provide upper ad lower bouds o log MP i terms of log P 0. I both cases, our results costitute improvemets to.7 whe P Z[z] ad P 0 is sufficietly small. Our first result is the lower boud. =3

4 88 K.-K. S. CHOI AND C. L. SAMUELS Theorem.. If P C[z] is such that P 0 =, the log P 0 log MP. If P has iteger coefficiets, P 0 0, ad P 0 <, the the hypotheses of Theorem. are automatically satisfied. Ideed, it follows from.6 that 0 < P 0 < for all P C[z]. Therefore, if P Z[z], the P 0 =. Furthermore, if MP e the log MP log MP, so that Theorem. does ideed provide a improvemet to the upper boud i.7. Pritser further otes that the aalogue of Lehmer s cojecture is false for P 0, providig the example P z=z.here,wehavethat log log MP =log ad log P 0 = O. Now let ZP deote the umber of roots of P that lie iside the ope uit dis. I this example, ad i aother provided by Pritser, ZP teds to as. For the remaider of this article, we will always assume that ZP. Otherwise Pritser s wor [9] establishes that MP = P 0 = P 0 ad there is little ew iformatio to discover. Our ext result gives a upper boud o log MP /ZP i terms of P 0..8 Theorem.3. If P C[z] is such that P 0 = ad P 0 <e the log MP ZP log P 0 + = + l + l log / P 0. + We have trivially that 0 /l l so that the iterior sum o the right had side of.8 coverges. Moreover, if we are willig to sacrifice a small amout of sharpess, we obtai a more cocrete result. Corollary.4. If P C[z] is such that P 0 = ad P 0 <e the log MP.9 log P 0 + log P ZP 3 0. log P 0 The right-had term i.9 should be regarded as a error term as P 0, showig that log MP /ZP is asymptotically bouded above by log P 0. Pritser s aforemetioed example establishes that the mai term i this estimate is best possible. Ideed, taig P z =z, we see that MP =ZP =, implyig that the left-had side of.8 equals log /. We also observe that log P 0 =log + / =log + log exp.

5 THE AREAL MAHLER MEASURE 89 Sice log exp = we obtai that = log! log P 0 = log = log log 3 + O + log. log 3 + O I other words, we have show that log MP log.0 =log P 0 +O. ZP Therefore, the mai term i.9 provides the best possible upper boud. We must acowledge, however, that improvemets to the error term still seem liely. I this example, Theorem.3 gives log MP ZP log log P 0 +O, the right-had side of which has a larger error term tha that of.0. We also ote that all roots of P = z that lie iside the uit circle have the same absolute value. For a geeral polyomial P C[z] satisfyig this property, we will say that P is balaced. I this case, we ca obtai a lower boud o log MP aalogous to Theorem.3. Theorem.5. If P is a balaced polyomial over C such that P 0 = ad P 0 <e the log MP log P 0. ZP Applyig Corollary.4 ad Theorem.5 to ay sequece of balaced polyomials P z with P 0, we fid that log MP ZP =log P 0 +O log P 0. I the case of P z=z, this formula becomes log MP log =log P 0 +O, ZP yieldig a asymptotic formula havig the same mai term as.0. Although we caot obtai the best error term i this specific example, the advatage of Theorems.3 ad.5 is that they apply far more geerally. 3,

6 830 K.-K. S. CHOI AND C. L. SAMUELS. Proofs For this sectio, we will always write fz =z / log z ad ote that. log P 0 = f z, provided that P 0 = ad z are the roots of P. By applyig the power series expasio for the logarithm, we fid that. fz= z z + =3 holds for all z <. Moreover, it ca be easily verified that.3 log z = fz+ l l z =3 i the same dis. This iformatio aloe is eough to prove Theorem.. Proof of Theorem.. We assume that N P z= a z = a N =0 N = z z, where a 0 = ad observe immediately from.5 that log MP = log z log z because log z > 0for z <. If 0 <x<, the it is clear from.3 that log x fx, which gives log MP f z =log P 0. I our proof of Theorem.3, it is importat to have a lower boud o the roots of P i terms of the areal Mahler measure. If P 0 is small, the Theorem.3a of [9] shows the roots of P are close to the uit circle. Our ext lemma gives a quatitative result i this directio. Lemma.. Suppose that P z=a N N = z z C[z] where P 0 = ad P 0 <e. The z log P 0 for all.

7 THE AREAL MAHLER MEASURE 83 Proof. Let fx=x / log x. Sicefx > 0 is decreasig o 0,, we fid from. that.4 log P 0 = f z { max f z } = f mi z. By., we fid that fx x for all x 0,. Combiig this with the above iequality, we obtai that completig the proof. mi z log P 0 Assume that P N C[z] is ay sequece of polyomials of degree N, each havig roots z N,,z N,,...,z N,N. If lim N log P N 0 =0, the Pritser showed i Theorem.3a of [9] that lim if N mi z N,. N The special case ivolvig sequeces P N with P N 0 = is a cosequece of Lemma.. Now we preset our proof of Theorem.3. Proof of Theorem.3. Applyig Cauchy s iequality, we obtai immediately that.5 log MP ZP log z. By.3, we see that log z = f z.6 + =log P 0 + =log P 0 + = =3 + =3 l l z l l z z l + l + z. Now let g :0, R be give by + gx= l + l x. + =

8 83 K.-K. S. CHOI AND C. L. SAMUELS Sice g x < 0 for all x 0,, we ow that g is decreasig o this iterval, ad Lemma. implies that g z g log P 0. Usig this observatio i.6, we see that log z log P 0 + g log P 0 log P 0 + g log P 0 =log P 0 + g log P 0 log P 0 ad the result follows from.5. z z z + Although the proof of Theorem.3 uses several estimates, we believe the weaest of these are the use of Cauchy s Iequality ad Lemma.. Our applicatio of Cauchy s Iequality i.5 admits equality wheever the values z are all equal i.e., wheever P is balaced. O the other had, iequality.4 i the proof of Lemma. is sharpest whe oe root of P is somewhat far from the uit circle while the others are close. To summarize, these estimates are sharpest i opposig cases, so that their simultaeous use leads to a weaer estimate that should be expected. Nevertheless, Lemma. is used oly to estimate the error term i Theorem.3. I Pritser s sequece P z=z, all roots of P have the same absolute value, meaig that we have equality i.5. Ideed, we foud that log MP log =log P 0 +O, ZP while Theorem.3 gives log MP log log P 0 +O. ZP As expected, these estimates have the same mai term, while differig i their error terms. Proof of Corollary.4. Let φx= x lx l x so we must show that φx /3 for all itegers x 3. We verify easily that φ3 = φ4 = /3 soit is eough to show that φx /3 for all x 5. Set ψx= x + x 3 x x ad observe that φx ψx for all x 3. We have that ψ5 = 9/30 < /3, ad it is a stadard calculus problem to show that ψx is decreasig o [5,. We ow obtai that φx /3 for all itegers x 3. =3

9 THE AREAL MAHLER MEASURE 833 Therefore, Theorem.3 yields log MP ZP log P log / P 0 = ad the result follows by summig the geometric series. Proof of Theorem.5. Sice all roots of P have the same absolute value, we obtai equality i.5 givig log MP =ZP log z. Therefore, we may apply.6 to obtai that log MP ZP =log P 0 + z It is straightforward to verify that + for all, so the result follows. = + l + l + 0 Refereces l + l z. + [] F. Amoroso ad R. Dvoricich, A lower boud for the height i abelia extesios, J. Number Theory , o., MR [] P. Borwei, E. Dobrowolsi ad M. J. Mossighoff, Lehmer s problem for polyomials with odd coefficiets, A. of Math , o., MR [3] E. Dobrowolsi, O a questio of Lehmer ad the umber of irreducible factors of a polyomial, Acta Arith , o. 4, MR [4] A. Dubicas ad C. J. Smyth, O metric heights, Period. Math. Hugar , o., MR [5] A. Dubicas ad C. J. Smyth, O the metric Mahler measure, J. Number Theory 86 00, MR 839 [6] P. Fili ad C. L. Samuels, O the o-archimedea metric Mahler measure, J.Number Theory 9 009, o. 7, MR 5490 [7] H. Huag, Mahler s Measures o fuctio spaces, preprit. [8] D. H. Lehmer, Factorizatios of certai cyclotomic fuctios, A. of Math , MR 5038 [9] I. E. Pritser, A areal aalog of Mahler s measure, Illiois J. Math , MR 5464 [0] C. L. Samuels, A collectio of metric Mahler measures, J. Ramauja Math. Soc. 5 00, o. 4, MR [] C. L. Samuels, The parametrized family of metric Mahler measures, J. Number Theory 3 0, o. 6, MR 77489

10 834 K.-K. S. CHOI AND C. L. SAMUELS [] A. Schizel, O the product of the cojugates outside the uit circle of a algebraic umber, Acta Arith , Addedum, ibid , o. 3, MR [3] C. D. Siclair, The rage of multiplicative fuctios o C[x], R[x] ad Z[x], Proc. Lodo Math. Soc , o. 3, MR [4] C. J. Smyth, O the product of the cojugates outside the uit circle of a algebraic iteger, Bull. Lodo Math. Soc. 3 97, MR Kwo-Kwog Stephe Choi, Simo Fraser Uiversity, Departmet of Mathematics, 8888 Uiversity Drive, Buraby, BC V5A S6, Caada address: choi@math.sfu.ca Charles L. Samuels, Olahoma City Uiversity, Departmet of Mathematics, 50 N. Blacwelder, Olahoma City, OK 7306, USA address: clsamuels@ocu.edu