IOSR Joual of haacy ad Biological Scieces (IOSR-JBS) e-issn:78-8, p-issn:19-7676 Volue 11, Issue Ve III (ay - Ju16), 1-18 wwwiosjoualsog Stog Result fo Level Cossigs of Rado olyoials 1 DKisha, AK asigh 1 Dept of atheatics ad Huaities, CET, BBSR, ODISHA, INDIA Dept of Basic Sciece ad huaities, IT, BBSR, ODISHA, INDIA Abstact: Let N be the ube of eal oots of the algebaic equatio f () whee ae idepedet ado vaiables assuig eal values oly The thee eists a itege such that fo each > the ube of eal oots of ost of the equatios f()= is at least e log ecept fo a set of easue at ost log 1991 atheatics subject classificatio (Ae ath Soc): 6 B 99 Keywods ad hases: Idepedet idetically distibuted ado vaiables, ado algebaic polyoial, ado algebaic equatio, eal oots I Theoe Let f (,w) be a polyoial of degee whose coefficiets ae idepedet ado vaiables with a coo chaacteistics fuctio ep C t, whee =1 ad C is a positive costat Tae, {e }to be ay sequece tedig to zeo such that e log teds to ifiity as teds to ifiity The thee eists a itege such that fo each > the ube of eal oots of ost of the equatios f()= is at least e log ecept fo a set of easue at ost μ log II Itoductio Let N be the ube of eal oots of the algebaic equatio whee f () DOI: 1979/8-11118 wwwiosjoualsog 1 age μ ae idepedet ado vaiables assuig eal values oly Seveal authos have estiated bouds fo N whe the ado vaiables satisfy diffeet distibutio laws Littlewood ad Offod [] ade the fist attept i this diectio They cosideed the cases whe the ae oally distibuted o uifoly distibuted i (-1, +1) o assue oly the values +1 ad 1 with equal pobability They obtaied i each case that N μlog logloglog 1 Saal [] has cosideed the geeal case whe the A log have idetical distibutio, with eceptio zeo, vaiace ad thid absolute oet fiite ad o-zeo He has show that N >s log outside a eceptioal set whose easue teds to zeo as teds to ifiity, whee s teds to zeo, but s log teds to ifiity Saal ad isha [4] have cosideed the case the C t whee C is a positive costat ad 1 They have show that N μlog loglog have a coo chaacteistics fuctio ep
Stog Result fo Level Cossigs of Rado olyoials outside a eceptioal set easue at ost ' μ (loglog)(log) μ loglog log 1, if1,if I all the above cases the eceptioal set depeds upo Evas [1], was the fist to obtai stog esult fo these bouds I such case the eceptioal set is idepedet of the degee of the polyoial We use the te stog esult i the followig sese: All the above esults ae of the fo N μ 1 as i teds to ifiity wheeas the theoe of Evas is of the fo sup N μ 1 as teds to ifiity Evas [1] has show, i case of oally distibuted coefficiets, that thee eists a itege such that fo > N μlog loglog ecept fo a set of easue at ost μloglog log Saal ad isha [5] have show i the case of chaacteistic fuctio ep N μlog loglog outside a eceptioal set of easue at ost μ log log loglog whee >1 1 C t that fo > I [7], they have cosideed the stog esult i the geeal case Assuig that the ado vaiables (ot ecessaily idetically distibuted) have eceptio zeo, vaiace ad thid absolute oet o-zeo fiite, they have show that fo N (μlog)/log (K outside a set of easue at ost μ log log K log log t /t )log DOI: 1979/8-11118 wwwiosjoualsog 1 age
K = a povided σ espectively of,t li t is fiite ad log aσ ad a τ σ,τ Stog Result fo Level Cossigs of Rado olyoials K t log = (log) whee beig the vaiace ad thid absolute oet Ou object is to ipove the stog esult fo lowe boud i case of chaacteistic fuctio ep C t We have show that fo > N > log μ Outside a eceptioal set of easue at ost log,but log whee The esult of Evas [1] is a special case of ous ad is obtaied by taig = ad 1 (loglog) i ou theoe 1 The esult of Saal ad isha [5] is also a special case of ou theoe 1 O the othe had ou eceptioal set is salle All authos who have estiated bouds fo N have used oe id of basic techique oigially used by Littlewood ad Offod [] We shall deote μ fo positive costats which ay have diffeet values i diffeet occueces We suppose always that is lage so that ay iequalities tue whe is lage ay be tae as satisfied Thoughout the pape, [] will deote the geatest itege ot eceedig It ay be oted that although Evas [1] is a special case of ous, a uch bette estiate fo the lowe boud with salle eceptioal set ca be deived fo ou theoe 1 Fo eaple, if we tae =, p (loglog whee <p<1, the fo > ) log N (loglog) p outside a eceptioal set of easue at ost μ(loglog ) log p Lea 1 If a ado vaiable ζ has chaacteistic fuctio ep C t, the fo evey e> C 1 1 1 This lea is due to Saal ad isha [4] oof of the Theoe Tae costat A ad B such that <B<1 ad A>1 Choose β such that β ad teds to ifiity Let So log logβ Ae / β, β 1 B (11) μ β μ 1 β We defie (X) log both ted to ifiity as DOI: 1979/8-11118 wwwiosjoualsog 14 age
Stog Result fo Level Cossigs of Rado olyoials Let be the itege deteied by 87 811 υ(8 7) υ(8 11) (1) The fist iequality gives llog μ log The secod iequality gives logβ log 8 11 (8 11)log(8 11) (8 11)log log (8 11) (8 11) So Thus f () log log (8 11)log log logβ μ μ μ log logβ μ log logβ (1) By the coditio iposed o β it follows that teds to ifiity as teds to ifiity We have at the poits U R X 1/ 1 1 4 υ(4 1) 1, /, (14) fo = / w hee U X,R X 1 41 υ(4 1) the ide v agig fo 1 4 υ(4 1) ad fo υ(4 ) 1 i 41 DOI: 1979/8-11118 wwwiosjoualsog 15 age 4 to υ(4 ) i, fo to to We also have f( ) UR,f( 1) U 1R 1 (15) Obviously U ad U +1 ae idepedet ado vaiables Agai it follows fo (1) that +1< 87 fo lage Also the aiu ide i U +1 fo = is υ(8 7), which, by (1) is cosistet with (15) Let V whe is lage 1/ The 1 υ(4 1) 4 1 1 V 1 υ(4 1) 4 1 1 4 4 4-1 (41) υ(4 1) υ(4-1) υ(4-1) 1 υ(4 1) 4 1 1 ( / ) e A υ(4 1) υ(4 4 1) 1 ( B / A) e (16) 1
Now we estiate U V Stog Result fo Level Cossigs of Rado olyoials, U 1 V1 U V, U 1V 1 U V, ( U V U V ) ( U V 1 1 Sice the chaacteistic fuctio of is ep C t C t ep C t V DOI: 1979/8-11118 wwwiosjoualsog 16 age 1 1, the chaacteistic fuctio of U is theefoe ep 4 1 8 whee the ide V ages fo υ(8 1) 1 to υ(8 ) i Theefoe the 4 U /V is ep C t, which is siilaly also the chaacteistic fuctio chaacteistic fuctio of U 1/V 1 Thus the chaacteistic fuctio is depedet o Let F() be the coo distibutio fuctio Hece U V ) ( U /V 1 1 ( U /V 1) 1 F(1) Thus =1 F(1) F( 1) F( 1) 1 F(1) F( 1) 1 F(1) δ(say) Obviously δ> 1 We shall eed the followig leas Lea 1 oof But Sice We have V 1 CAe ep - (4 1) B(1 ) The chaacteistics fuctio of is ep C t 1 (1 )V C / ecept fo a set of easue at ost υ(4) υ(4 ) υ(4 X 41 41 ) 4 1 > > > X fo sufficietly lage υ(4) 1 1 1 υ(4 ) (4 1) υ(4 1) Hece usig (16), we obtai as equied (4 ) υ(4 1)(4 4 41 41 υ(4) 41 log(4 ) (4) 4 log(41) (41) 41 4 1) ep (4 1) 1 CAe 1 ep (4 1) B(1 )
Stog Result fo Level Cossigs of Rado olyoials Lea 1 1/ This follows diectly fo lea 11 1 Now 1/ 1/ 1/ 1 V 1/ υ(4 1) (4 1) 41 υ(4 1) 41 1/ [LOG(41) (41) (4 1) (4 1) Ae V B 16 υ(4 1) [LOG(41) (41) ecept fo a set of easue at ost 41 1 1 υ(4 1) 41 4 16 υ (4 1) 16 Ae V B 4 4 4 1/ 1/ 1/ β Ae V B 1/ 1 1 1/ 41 1/ 1/ 1/ 1 C (1 ) The last two steps above follow fo (11) ad (16) Hece by usig leas 1 ad 1, we have R < V fo evey sufficietly lage ecept fo a set of easue at ost μep Thus we have R μ' (4 1) μep ( ) V ad R 1 V 1 fo =, +1, whee =[/]+1 The easue of the eceptioal set is at ost μ' DOI: 1979/8-11118 wwwiosjoualsog 17 age
ep ep ' (4 ep ( 1) ) Stog Result fo Level Cossigs of Rado olyoials ' 1 ( ) (17) 14 We defie the evets E ad F as follows: E ={U >V, U +1 <-V +1 } F ={U <V, U +1 >-V +1 } We have show ealie that (E F ) δ Let η be a ado vaiable such that it taes value 1 o E Uf ad zeo elsewhee I othe wods η 1, with pobabilityδ, with pobability 1- δ Let η ae thus idepedet ado vaiables with E (η )=δ ad V (η )= δ δ <1 We wite S if R Vad R 1 V 1 1othewise By cosideig the polyoial whee f () III Coclusio ae idepedet ado vaiables assuig eal values oly we foud that the ube of zeos of the above polyoial of the equatios f()= is at least (e log ) ecept fo a set of easue at ost itege > the ube of eal oots of ost log fo a Refeeces [1] JE Littlewood ad AC Offod, O the ube of eal oots of a ado algebaic equatio II, oc Cab hilos Soc, 5(199), 1-148 [] J Haje ad A Reye, A geealizatio of a iequality of Kologoov, Acta ath Acad Sci Hugay, 6(1955), 81-8 [] GSaal, O the ube of a ado algebaicequatio, oc Cab hilos Soc, 58(196), 4-44 [4] EA Evas, O the ube of a ado algebaicequatio, oc Lodo ath Soc, (15)(1965), 71-749 [5] GSaal ad D atihai, Stog esult fo eal zeos of ado polyoials, JIdia ath Soc, 4(1976), -4 [6] GSaal ad D atihai, Stog esult fo eal zeos of ado polyoials, II JIdia ath Soc, 41(1977), 95-4 [7] N Ragaatha ad Sabadha, O the lowe boud of the ube of eal oots of a ado algebaic equatio, Idia ue Appl ath, 1(198), 148-157 [8] NNNaya ad S ohaty, O the lowe boud of the ube of eal zeos of a ado algebaic polyoial, J Idia ath Soc, 49(1985), 7-15 DOI: 1979/8-11118 wwwiosjoualsog 18 age