Non-Archimedean Analysis on the Extended Hyperreal Line d and the Solution of Some Very Old Transcendence Conjectures over the Field

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1 Avaces i Pue atheatics Pubishe Oie August 25 i SciRes htt://wwwsciog/joua/a htt://xoiog/4236/a25556 No-Achieea Aaysis o the Extee Hyeea Lie a the Soutio of Soe Vey O Tasceece Cojectues ove the Fie Jayov Fouzo Cete fo atheatica Scieces Isae Istitute of Techoogy Haifa Isae Eai: jayovfouzo@istu Receive 9 ay 25; accete 5 August 25; ubishe 9 August 25 Coyight 25 by autho a Scietific Reseach Pubishig Ic This wo is icese ue the Ceative Coos Attibutio Iteatioa Licese (CC BY htt://ceativecoosog/iceses/by/4/ Abstact I 98 F Wattebeg costucte the Deei coetio of the Robiso o-achie- ea fie a estabishe basic agebaic oeties of I 985 H Gosho estabishe futhe fuaeta oeties of I [4] iotat costuctio of suatio of coutabe sequece of Wattebeg ubes was oose a coesoig basic oeties of such suatio wee cosiee I this ae the iotat aicatios of the Deei coetio i tasceeta ube theoy wee cosiee Give ay aaytic fuctio of f z at oe coex vaiabe f [ z] [ ] we ivestigate the aithetic atue of the vaues of ( tasceeta oits e ai esuts ae: the both ubes e + π a e π ae iatioa; 2 ube e is tasceeta Notivia geeaizatio of the Liea-Weiestass e theoe is obtaie Keywos No-Achieea Aaysis Robiso No-Achieia Fie Deei Coetio Deei Hyeeas Wattebeg Ebeig Gosho Ieotet Theoy Gosho Tasfe How to cite this ae: Fouzo J (25 No-Achieea Aaysis o the Extee Hyeea Lie a the Soutio of Soe Vey O Tasceece Cojectues ove the Fie Avaces i Pue atheatics htt://xoiog/4236/a25556

2 J Fouzo Itouctio I 873 Fech atheaticia Chaes Heite ove that e is tasceeta Coig as it i yeas afte Eue ha estabishe the sigificace of e this eat that the issue of tasceece was oe atheaticias cou ot affo to igoe Withi yeas of Heite s beathough his techiques ha bee extee by Liea a use to a π to the ist of ow tasceeta ubes atheaticia the tie to ove that othe ubes such as e + π a e π ae tasceeta too but these questios wee too ifficut a so o futhe exaes eege ti toay s tie The tasceece of e π has bee ove i 929 by A O Ge fo Cojectue Whethe the both ubes e + π a e π ae iatioa Cojectue 2 Whethe the ubes e a π ae agebaicay ieeet Howeve the sae questio with e π a π has bee aswee: Theoe (Nesteeo 996 [] The ubes e π a π ae agebaicay ieeet Thoughout of 2-th cetuya tyica questio: whethe f ( α is a tasceeta ube fo each agebaic ube α has bee ivestigate a aswee ay authos oe esut i the case of etie fuctios satisfyig a iea iffeetia equatio ovies the stogest esuts eate with Siege s E-fuctios [] [2] ef [] cotais efeeces to the subject befoe 998 icuig Siege E a G fuctios Theoe (Siege C L Suose that λ λ 2 α ϕ λ ( z z ( ( λ+ ( λ+ 2 ( λ+ The ϕλ ( α is a tasceeta ube fo each agebaic ube α Let f be a aaytic fuctio of oe coex vaiabe f [ z] Cojectue 3 Whethe f ( α is a iatioa ube fo give tasceeta ube α Cojectue 4 Whethe ( I this ae we ivestigate the aithetic atue of the vaues of ( Defiitio Let g( x : be ay ea aaytic fuctio such that f α is a tasceeta ube fo give tasceeta ube α f z at tasceeta oits e ( < [ ] g x ax x a (2 We wi ca ay fuctio give by Equatio (2 -aaytic fuctio a eote by g ( x Defiitio 2 [3] [4] A tasceeta ube z is cae -tasceeta ube ove the fie g x such that g ( z ie fo evey -aaytic fuctio g ( x the iequaity g ( z is satisfies Defiitio 3 [3] [4] A tasceeta ube z is cae w-tasceeta ube ove the fie if if thee oes ot exist -aaytic fuctio ( z is ot -tasceeta ube ove the fie ie thee exists -aaytic fuctio g ( x such that g ( z Exae Nube π is tasceeta but ube π is ot -tasceeta ube ove the fie as ( fuctio si x is a -aaytic a π (2 si ie ( π + ( π π π π ( ! 2 5! 2 7! 2 2! ai esuts ae Theoe [3] [4] Nube e is -tasceeta ove the fie Fo theoe ieiatey foows Theoe 2 Nube e e is tasceeta Theoe 3 [3] [4] The both ubes e + π a e π ae iatioa 588

3 J Fouzo Theoe 4 Fo ay ξ ube e ξ is -tasceeta ove the fie Theoe 5 [3] [4] The both ubes e π a e π ae iatioa Theoe 6 [3] [4] Let f ( z 2 be a oyoias with coefficiets i Assue that fo ay agebaic ubes ove the fie : β β 2 fo a coete set of the oots of f ( z such that a a 2 ; a Assue that The ( [ ]eg ( f z z f z (4 + < a a e β (5 + a a e β (6 2 Peiiaies Shot Outie of Deei Hyeeas a Gosho Ieotet Theoy Let be the set of ea ubes a a ostaa oe of [5] is ot Deei coete Fo exae µ ( { x x } a ae boue subsets of which have o suea o ifia i Possibe coetio of the fie ca be costucte by Deei sectios [6] [7] I [6] Wattebeg costucte the Deei coetio of a ostaa oe of the ea ubes a aie the costuctio to obtai cetai is of secia easues o the set of iteges Thus was estabishe that the Deei co- etio of the fie is a stuctue of iteest ot fo its ow sae oy a we estabish futhe iotat aicatios hee Iotat cocet itouce by Gosho [7] is that of the absotio ube of a ee- et a which oughy seaig easues the egee to which the caceatio aw a+ b a + c b c fais fo a 2 The Deei Hyeeas Defiitio 2 Let be a ostaa oe of [5] a P ( the owe set of A Deei hyeea α is a oee ai { UV } P( P( coitios: x y( x U y V 2 U V 3 x( x U y( y V x< y 4 x( x V y( y V x< y 5 x y( x< y x U y V Coae the Defiitio 2 with oigia Wattebeg efiitio [6](see [6] efii Desigatio 2 Let { } UV α We esigate i this ae U cut ( α V cut+ ( α { cut ( cut+ ( } α α α Desigatio 22 Let α We esigate i this ae { } ( cut ( α cut α α + α α α α is the set: { x x α} that satisfies the ext Rea 2 The oa of α is eote by µ ( α Sueu of µ ( is eote by ε Sueu of is eote by Note that [6] 589

4 J Fouzo ( ] ( ( ε µ Let A be a subset of boue above The ( Exae 2 su ( + \ 2 ε ( µ ( su A exists i [6] su \ Rea 22 Ufotuatey the set iheits soe but by o eas a of the agebaic stuctue o Fo exae is ot a gou with esect to aitio sice if x+ y eotes the aitio i the: ε + ε ε + ε Thus is ot eve a ig but seuo-ig oy Defiitio 22 We efie: The aitive ietity (zeo cut 2 The utiicative ietity ofte eote by { x x< } ofte eote by { x x< } o siy is o siy is Give two Deei hyeea ubes α a β we efie: 3 Aitio α + β of α a β ofte eote by α + β is { x yx y } α + β + α β It is easy to see that α + α fo a α It is easy to see that α + β is agai a cut i a α + β β + α Aothe fuaeta oety of cut aitio is associativity: ( α + β γ α ( β γ This foows fo the coesoig oety of 4 The oosite α of α ofte eote by ( α o siy by α is { x x x } α α is ot the east eeet of \ α 5 We say that the cut α is ositive if < α o egative if α < The absoute vaue of α eote α is α α if α a α α if α 6 If αβ the utiicatio α β of α a β ofte eote α β is { z z x y x y xy } α β fo soe α β with I geea α β if α o β α β α β if α β o α < β < α β α β if α β < o α < β ( 7 The cut oe ejoys o the staa aitioa oeties of: (i tasitivity: α β γ α γ (ii tichotoy: eize α < β β < α o α β but oy oe of the thee (iii tasatio: α β α + γ β + γ 22 The Wattebeg Ebeig ito Defiitio 23 [6] Wattebeg hyeea o -hyeea is a oety subset α such that: (i Fo evey a α a b< a b α 59

5 J Fouzo (ii α α (iii α has o geatest eeet Defiitio 24 [6] I ae [6] Wattebeg ebe ito by foowig way: If α the coesoig eeet α of is { x x α} α < (2 Rea 23 [6] I ae [6] Wattebeg oite out that coitio (iii above is icue oy to avoi ouiqueess Without it α wou be eesete by both α a α { α} Rea 24 [7] Howeve i ae [7] H Gosho oite out that the efiitio (2 i Wattebeg ae [6] is techicay icoect Note that Wattebeg [6] efies α i geea by α { a a α} (22 If α ie \α has o iiu the thee is o ay obe with efiitios (2 a (22 Howeve if α a fo soe a wheeas the efiitio of equies that: ie { x x < a} α the accoig to the atte efiitio (22 { x x a} α < (23 { x x a} < α (24 but this is a cotaictio Rea 25 Note that i the usua teatet of Deei cuts fo the oiay ea ubes both of the atte sets ae egae as equivaet so that o seious obe aises [7] Rea 26 H Gosho [7] efies α by { x bb [ a b a] } α (25 Defiitio 25 (Wattebeg ebeig We ebe ito of the foowig way: (i if α the coesoig eeet α of is a α { x x α} (26 { a a } { } α α α (27 o i the equivaet wayie if α the coesoig eeet α of is Thus if α the α A B whee { α} α x x (28 { α} { α} A x x B y y (29 Such ebeig ito Such ebeig we wi ae Wattebeg ebeig a to esigate by Lea 2 [6] (i Aitio ( + (ii α : α + α is coutative a associative i (iii ( αβ : α + β α+ β Rea 27 Notice hee agai soethig is ost goig fo to sice a < β oes ot iy α + α < β + α sice < ε but + ε ε + ε ε 59

6 J Fouzo Lea 22 [6] (i a iea oeig o ofte eote which extes the usua oeig o + + (ii ( α α ( β β α β α β (iii α < α β β α β α β < + < + (iv is ese i That is if α < (v Suose that A (vi Suose that A β i thee is a a the α is boue above the su A suα α A α A ( α is boue beow the if A if α α A α A + ( α Rea 28 Note that i geea case α ( α < a < cut exist i cut exist i β if if cut I aticua the foua fo if A α A α A A give i [6] o the to of age 229 is ot quite coect [7] see Exae 22 Howeve by Lea 22 (vi this is o obe if if cut α says α A α A A Exae 22 [7] The foua ( if a a α A + cut α A ( ( α Let A be the set A { a+ } whee us though the set of a ositive ubes i the A a { xx< a} Howeve ( α { xxa} if Lea 23 [6] (i If α cut α A the ( α ( α α α (ii ( (iii α β β α + + (iv ( α β α β (v a ( a + β ( a β + : (vi α + ( α Poof (v By (iv: ( a + ( β ( a + β ( Suose ow c ( a + β this eas (2 c c< c ( a + β a theefoe (3 c ( a + β (4 Note that: ca β (sice c a β a c c a but this is a cotaictio (5 Thus ca β a theefoe c+ a β (6 By siia easoig oe obtais: c + a β (7 Note that: a( c c a a theefoe c a c c + ca a + β a ( iy ( ( ( ( ( ( c a c c + c + a a + β 592

7 J Fouzo Lea 24 (i a β µ µ ( µ a + ( µβ µ ( a + β (ii a β µ µ : ( µ a µβ µ ( a + + β : Poof (i Fo µ the stateet is cea Suose ow without oss of geeaity µ By Lea 23 µ a + µβ µ a + µβ c ( Suose c µ ( a + β a theefoe ( a + β but this eas µ (iv: ( ( ( c < c µ µ c a + β µ (2 c ( a + β (3 ( a theefoe c c (4 Note that: a β (sice a β a µ µ c c c c + + µ µ µ µ c (5 Thus a β a theefoe µ a a a β (6 By siia easoig oe obtais: (7 Note that: ( c c a a µ µ but this is a cotaictio c+ c µ a µβ + µ µβ a µ a c c µ a a theefoe ( ( ( ( c µ a c c + c + µ a µ a + µβ iy (ii Ieiatey foows fo (i by Lea 23 Defiitio 26 Suose α The absoute vaue of α witte α is efie as foows: α if α α α if α Defiitio 27 Suose αβ The ouct α β is efie as foows: Case ( αβ { ( ( } a b a b ( α β < < α < < β Case (2 α β : α β Case (3 ( α < ( β < ( α < β < : (2 α β α β iff α < β < α β ( α β iff ( α < ( β < (2 Lea 25 [6] (i ( (ii utiicatio ( ab : a b a b is associative a coutative: (iii α α; ( α β γ α ( β γ α β β α α α whee ( (22 593

8 J Fouzo (iv α β β α α β γ α β + γ α β + α γ (v ( ( ( ( (vi < α < α < β < β α β < α β Lea 26 Suose µ a βγ The ( ( ( µ β µ β γ µ β µ γ Poof We choose ow: ( a such that: γ + a (2 Note that µ ( β γ µ ( β γ + µ a µ a The fo (2 by Lea 24 (ii oe obtais (3 µ ( β γ µ ( β γ + a µ a Theefoe (4 µ ( β γ µ β ( a γ + µ a (5 The fo (4 by Lea 25 (v oe obtais µ β γ µ β + µ a γ µ a (6 ( ( The fo (6 by Lea 24 (ii oe obtais µ β γ µ β + µ a µγ µ a µ β µγ (7 ( Defiitio 28 Suose α α the < (i < α α { a a α } : if (ii ( α : α α < Lea 27 [6] (i ( ( a : a a w α is efie as foows: (23 (24 (25 (ii (iii α α < α β β α (iv ( < α ( < β ( α β α β (v ( ( (vi a : a a β a β α α Lea 28 [6] Suose that a a βγ β γ The ( a β + γ a β + a γ Theoe 2 Suose that S is a o-ety subset of which boue fo above ie su( S exist a suose that ξ ξ The 594

9 J Fouzo { ξ x} ξ { x } ξ ( su su su S (26 x S x S Poof Let B su S The B is the saest ube such that fo ay x x T ξ xx S Sice ξ ξ x ξ B fo ay x S Hece T is boue above by ξ B Hece T has a sueu C T s su T Now we have to ove that CT ξ B ξ ( su S Sice ξ B ξ ( sus is a ue bou fo T a C is the saest ue bou fo T CT ξ B Now we eeat the aguet above with the oes of S a T evese We ow that C T is the saest ube such that fo ay y T y C T Sice S {( ξ yy T } Hece ( ξ B ( ξ C T a B C ξ B C T 23 Absotio Nubes i ξ it foows that ( ξ y ( ξ S B Let { } C T fo ay y T But C T is a ue bou fo S But B is a sueu fo S Hece ξ T We have show that C T ξ B a aso that B C ξ T Thus Oe of staa ways of efiig the coetio of ivoves estictig oesef to subsets which have the foowig oety ε ε xx α yy α [ y x< ε ] It is we ow that i this case we obtai a fie I fact the oof is essetiay the sae as the oe use i the case of oiay Deei cuts i the eveoet of the staa ea ubes ε of couse oes ot have the above oety because o ifiitesia wosthis suggests the itouctio of the cocet of absotio at ab α ( of a ube α fo a eeet α of which oughy seaig easues how uch α eats fo havig the above oety [7] Defiitio 29 [7] Suose α the Exae 25 (i α : ab ( α (ii ab ( ε ε ab ε ε (iii ( (iv ab ( (v ab ( α : α + ε ε α : α ε ε Lea 29 [7] c α < c ab α ( α { xx α [ x+ α] } ab (27 (i < ab ( a ( (ii c ab ( α a ab ( α c+ ab ( α Rea 29 By Lea 27 α ( eeets of to ab α ( Of couse if the coitio i the efiitio of α ( autoaticay get a the egative eeets to be i ab α ( sice x y α x α efiitio is that the ea iteest ies i the o-egative ubes A techicaity occus if ab ( α { } the ietify ab α ( with [ ab α ( becoes { } Rea 2 By Lea 27(ii ab α ( is aitive ieotet ab ay be egae as a eeet of by aig o a egative ab is eete we < The easo fo ou We xx< which by ou eay covetio is ot i ] Lea 2 [7] (i ab α ( is the axiu eeet β such that α + β α (ii ab α ( α fo α (iii If α is ositive a ieotet the ab ( α α Lea 2 [7] Let α satsify α The the foowig ae equivaet I what foows assue ab 595

10 J Fouzo (i α is ieotet (ii ab α a+ b α (iii a α 2 a α (iv [ a α a α] (v a α a α fo a fiite Theoe 22 [7] ( α + α ab ( α Theoe 23 [7] ab ( α + β ab ( α Theoe 24 [7] α + β α + γ ab α + β γ (i ( (ii ab ( ab ( α + β α + γ α + β α + γ Theoe 25 [7] Suose αβ the ab α ab α (i ( ( (ii ab ( α + β ax { ab ( α ab ( β } Theoe 26 [7] Assue β If α absobs β the α absobs β Theoe 27 [7] Let < α The the foowig ae equivaet (i α is a ieotet α + α α (ii ( ( (iii ( α + α α (iv Let a 2 be two ositive ieotets such that 2 + The ( Gosho Tyes of α with Give ab(α Aog eeets of α such that ab α ( oe ca istiguish two ay iffeet tyes foowig [7] Defiitio 2 [7] Assue (i α has tye if x( x α y[ x+ y α y ] (ii α have tye (iii α (iv α has tye 2 if x( x α y( y [ x y α] + ie α has tye 2 iff α oes ot has tye A if x( x α y[ xy α y ] α α α has tye 2A if x( x y( y [ x y ] 25 Robiso Pat R{ α } of Absotio Nube α ( Theoe 28 [6] Suose α ( a at of α a eote by Wst ( α such that: (i ( x [ α ε α + ε ] (ii α β iies Wst ( α Wst ( β ( : (iv ( ( ( (iii The a Wst is cotiuous Wst α + β Wst α + Wst β Wst Wst Wst (v ( α β ( α ( β Wst α Wst α (vi ( ( (vii Wst ( α Wst ( α if α [ ε ε ] Theoe 29 [7] The thee is a uique staa x cae Wattebeg sta- 596

11 J Fouzo (i α has tye iff α has tye A (ii α caot have tye a tye A siutaeousy (iii Suose ab ( α The α has tye iff α has the fo ab α α a + fo soe a (iv Suose ( has tye A iff α has the fo ( (v If ab ( α ab ( β the α + β has tye iff α has tye (vi If ab ( α ab ( β the α + β has tye 2 iff eithe α o β has tye 2 Poof (iii Let α a + The ab α ( Sice a a a wite a as ( a + a + fo soe a + (we chose such that < It is cea that a wos to show that α has tye Covesey suose α has tye a choose a α such that: ya [ + y α y ] The we cai that: α a + By efiitio of ab α ( cetaiy a + α O the othe ha by choice of a evey eeet of α has the fo a+ with Choose such that the a+ a ( + a+ Hece α a + Theefoe α a + Exaes (i ε has tye a theefoe ε has tye A Note that aso ε has tye 2 (ii Suose ε ε The ε ε has tye a theefoe ε ε has tye A (ii Suose α ab ( α ε ie α has tye a theefoe by Theoe 29 α has the fo ( a + ε fo soe uique a a ( α tio ube α by foua (iii Suose ( fo ( a Wst The we efie uique Robiso at [ α ] R R { α} ( a { α} ( Wst ( α R of abso- (28 α ab α ε ie α has tye A a theefoe by Theoe 29 α has the ε fo soe uique a a ( α absotio ube α by foua a a (iv Suose α ( α The we efie Robiso at { α} Wst The we efie uique Robiso at [ α ] R R { α} ( a { α} ( Wst ( α R of (29 ab a α has tye A ie α has the fo a + fo soe R of absotio ube α by foua { α} a R (22 (v Suose α ab ( α a α has tye A ie α has the fo ( The we efie Robiso at R { α} of absotio ube α by foua { α} a a + fo soe R (22 Rea 2 Note that i geea case ie if α ( Robiso at { α} α is ot uique Rea 22 Suose α obtais the eesetatio a α ( R of absotio ube has tye o tye A The by efiitios above oe 597

12 J Fouzo { } ( α R α + ab α 26 The Pseuo-Rig of Wattebeg Hyeiteges Lea 22 [6] Suose that α The the foowig two coitios o α ae equivaet: (i α { ν ( ν ( ν α } (ii α if { ν ( ν ( α ν } su Defiitio 2 [6] If α satisfies the coitios etioe above α is sai to be the Wattebeg hyeitege The set of a Wattebeg hyeiteges is eote by Lea 23 [6] Suose αβ The (i α + β (ii α (iii α β The set of a ositive Wattebeg hyeiteges is cae the Wattebeg hyeatuas a is eote by Defiitio 22 Suose that (i λ ν (ii ˆ λ λ ˆ ν ν a (iii λ ν If ˆλ a ˆ ν satisfies these coitios the we say that ˆ ν is ivisibe by ˆλ a we eote this by λ ν Defiitio 23 Suose that (i α ( α su { ν ( ν ( λ ν ( ν α } (2 α { ν ( ν ( λ ν ( α ν } a (ii thee exists o if λ such that If α satisfies the coitios etioe above the we say that α is ivisibe by by λ α Theoe 2 (i Let ( α be a Wattebeg hyeatua such that (i (ii α has tye iff α has tye A (iii α caot have tye a tye A siutaeousy (iv Suose α ( α λ a we eote this be a ie hyeatuas such that (i ( α The ( ( + α ab The α has tye iff α has the fo a α a (v Suose α ab α α soe a α a (vi Suose α (vii Suose α ( If ( α ( β If ( α ( β Let a + fo soe has tye A iff α has the fo ( ab ab the α + β has tye iff α has tye a + fo ab ab the α + β has tye 2 iff eithe α o β has tye 2 Poof (i Ieiatey foows fo efiitios (22-(23 (iv Let α a + The ab α ( Sice a a+ (we chose such that < a wite a as ( a + It is cea that a wos to show that α has tye Covesey suose α has tye a choose a α such that: ya [ + y α y ] The we cai that: α a + By efiitio of ab α ( cetaiy a + α O the othe ha by choice of a evey eeet of α has the fo a+ with Choose such that the a+ a ( + a+ 598

13 J Fouzo Hece α a + Theefoe α a + 27 The Itege Pat It(α of Wattebeg Hyeeas α Defiitio 24 Suose α α Obviousy thee ae two ossibiities: A set { ν ( ν ( ν α } Poety I: [ α] Sice [ α] a + < α The we efie ( α [ α] [ α] { ν ( ν ( ν α } su It by foua has o geatest eeet I this case vai oy the α < α iies a cotaictig [ ] 2 A set { ν ( ν ( ν α } Poety II: [ α] υ such that [ ] α < a < a + α < a < α But the a < α has a geatest eeet υ I this case vai the a obviousy υ [ α] α < [ α] + υ+ α The we efie ( α [ α] fo α It ( α [ α] fo α < Defiitio 25 Suose Note that obviousy: It ( α It ( α It by foua which iies 28 Extea Su of the Coutabe Ifiite Seies i This subsectio cotais ey efiitios a oeties of su of coutabe sequece of Wattebeg hyeeas by Defiitio 2 6 [4] Let { } (i ( o (ii ( s s be a coutabe sequece s : such that s < o (iii { s } { } { } 2 ( ˆ s s s ( s The extea su (-su Theoe 2 (i Let { s } i s η Ext- s 2 2 ˆ ˆ ˆ 2 < 2 of the coesoig coutabe sequece s : ( i ( s : Ext-s su ( s ( ii ( s < : Ext-s if s su ( s ( iii ( s ( s < 2 2 : Ext- s Ext- s + Ext- s 2 ˆ ˆ 2 2 * The su ( (ii Let { s } The if ( * { } ( { } ( be a coutabe sequece s : s η ε be a coutabe sequece s : s η + ε is efie (222 such that ( [ s s ] a + such that ( [ s < s ] a i s η + 599

14 J Fouzo (iii Let { s } ifiite seies (iv Let { s } be a coutabe sequece s : s absoutey coveges to η i The such that ( [ s ] ( ( Ext- s su s η ε be a coutabe sequece s : a ifiite seies s absoutey coveges to η i The (v Let { s } s η < a (223 such that ( [ s < ] ( ( Ext- s if s η ε be a coutabe sequece s : s η + (224 such that s s s s s < ˆ ˆ 2 a ( { } { } { } ( ˆ 2( ˆ 2 2 (2 The s η < s η 2 2 ˆ ˆ ˆ ˆ 2 2 ( ( 2 Ext- s Ext- s + Ext- s η + η ε (225 Poof (i Let ( [ s s ] a i s η + Thus ε thee exists such that ( ( : η ε < s + < η Theefoe fo ( by Robiso tasfe oe obtais (2 ε : η ε < s + < η (2 ( ( ( ( Usig ow Wattebeg ebeig fo (2 we obtai (3 : s ε η ε < + < η (3 ( ( ( ( Fo (3 oe obtais (4 ε : η ε < su s < η (4 ( ( ( + ( Note that δ ( δ ( δ obviousy su s < η δ (5 ( ( Fo (4 a (5 oe obtais (6 s < η The obviousy: ( [ ] { } (6 ( ( ( ( ( ( s ( ε ε δ δ δ η ε < su < η δ Thus (i ieiatey fo (6 a fo efiitio of the ieotet Poof(ii Ieiatey fo (i by Lea 23 (v ε Poof(iii Let η s The obviousy: η < η a i η η Thus ε thee exists such that ( ( : η ε < η+ < η Theefoe fo ( by Robiso tasfe oe obtais (2 ε : η ε < η < η (2 ( ( ( + ( Usig ow Wattebeg ebeig fo (2 we obtai (3 ε : η ε < η < η (3 ( ( ( + ( Fo (3 oe obtais (4 6

15 (4 ε ( η ( ε < ( η+ < ( η : su Fo (4 by Defiitio 26 (i oe obtais ε : η ε < Ext- s < η (5 ( ( ( ( Note that δ ( δ ( δ (6 ( ( obviousy Ext- s < η δ Fo (5-(6 foows (7 (7 ( ( ( ( ( Ext ( s ( J Fouzo ε ε δ δ δ η ε < - < η δ Thus Equatio (223 ieiatey fo (7 a fo efiitio of the ieotet ε Poof(iv Ieiatey fo (iii by Lea 23 (v Poof(v Fo Defiitio 26(iii a Equatio (223-Equatio (224 by Theoe 27(iii oe obtais Theoe 22 Let { } ( ( ( ˆ ˆ 2 2 Ext- s Ext- s Ext- s η ε η ε a a absoutey coveges i Let a sequece { } the sequece { } Let { } ( ( 2 ( ( 2 η η ε ε η η ε b be a coutabe sequece a : a a ifiite seies s Ext- a be extea su of the coesoig coutabe such that ( be a coutabe sequece whee b a ( is ay eaageet of tes of a The extea su σ Ext- b has the sae vaue s as extea su of the coutabe sequece { a } Theoe 23 (i Let { } ifiite seies a of the coesoig coutabe sequece { } ie σ s ε be a coutabe sequece a : a absoutey coveges to η + i a et coesoig sequece { } (ii Let { } a a such that ( ( Ext- a The fo ay c + the equaity is satisfie ( η b a (2 be extea su of the c Ext- a Ext- c a c c ε (226 be a coutabe sequece a : a absoutey coveges to η i a et a sequece { } (iii Let { s } such that ( ( Ext- a The fo ay c + the equaity is satisfie: ( η a < (2 ifiite seies be extea su of the coesoig c Ext- a Ext- c a c c ε (227 be a coutabe sequece s : such that ( { } { } { } ( ( s s s s s < (2 ifiite seies (3 ifiite seies s absoutey coveges to η + i s absoutey coveges to η2 i 2 6

16 J Fouzo The the equaity is satisfie: c Ext- s Ext- c s + Ext- c s 2 ˆ ˆ 2 2 (( ( 2 c + η η c ε Poof (i Fo Defiitio 26 (i by Theoe 2 Theoe 2 (i a Lea (24 (ii oe obtais (( ( Ext- c a c Ext- a c η ε c η c ε (228 (ii Staightfowa fo Defiitio 26 (i a Theoe 2 Theoe 2 (ii a Lea (24 (ii oe obtais (( ( Ext- c a c Ext- a c η + ε c η + c ε (iii By Theoe 2 (iii a Lea (24 (ii oe obtais (( ( 2 ( ( 2 ( c Ext- s c η + η ε c η + η c ε But othe sie fo (i a (ii foows Defiitio 27 Let { a} Ext- c s + Ext- c s 2 ˆ ˆ 2 2 ( ( ( η ( η2 c ε c η c ε + c η + c ε c ( + absoutey coveges i to η ± We assue ow that: be a coutabe sequece a : such that ifiite seies (i thee exists such that : a η o (ii thee exists such that : a < η o (iii thee exists ifiite sequece i i 2 such that (a i : a i η a ifiite seies i a i absoutey coveges i to η a i (b thee exists ifiite sequece j j 2 such that j : a j < η a ifiite seies j a j j absoutey coveges i to η The: (i extea ue su (-ue su of the coesoig coutabe sequece a : is efie by ( Ext a ( a i - if ( ii Ext- a if ( a i i i i (ii extea owe su (-owe su of the coesoig coutabe sequece a : a (229 is efie by ( Ext a ( a i - su ( ii Ext- a su ( a j j j j (23 62

17 J Fouzo Theoe 24 ( Let { a} absoutey coveges i to η ± We assue ow that: be a coutabe sequece a : such that ifiite seies (i thee exists such that : a η o (ii thee exists such that : a < η o (iii thee exists ifiite sequece i i 2 such that (a i : a i η a ifiite seies i a i absoutey coveges i to η a i (b thee exists ifiite sequece j j 2 such that j : a j < η a ifiite seies j a j j absoutey coveges i to η a The ( ( η + ( ( η Ext-a if a ε Ext-a su a ε ( ( η + ( ( η j Ext-a if a ε i i i i Ext-a su a ε j j j Poof (i (ii (iii staightfowa fo efiitios Theoe 25 ( Let { } a absoutey coveges i to η ± We assue ow that: be a coutabe sequece a : such that ifiite seies a (23 (232 (i thee exists such that : a η o (ii thee exists such that : a < η o (iii thee exists ifiite sequece i i 2 such that (a i : a i η a ifiite seies i a i absoutey coveges i to η a i (b thee exists ifiite sequece j j 2 such that j : a j < η a ifiite seies j a j j absoutey coveges i to η The fo ay c the equaities is satisfie a + Ext- c a c Ext- a c ( η + c ε Ext- c a c Ext- c a c ( η c ε a (233 Ext- c a c - ( i Ext ai c η + c ε i i Ext- c a c Ext- a ( j c η c ε j j j Poof Coy the oof of the Theoe 23 (234 63

18 J Fouzo Theoe 26 ( Let { a} absoutey coveges i to η We assue ow that: be a coutabe sequece a : such that ifiite seies (i thee exists such that : a o (ii thee exists such that : a < o (iii thee exists ifiite sequece i i 2 such that (a i : a i a ifiite seies i a i absoutey coveges i to η a i (b thee exists ifiite sequece j j 2 such that j : a j < a ifiite seies j a j j absoutey coveges i to η a The fo ay c the equaities is satisfie + Ext- c a c Ext- a c ε Ext- c a c Ext- c a c ε Poof ( Fo Equatio (23 we obtai Ext- c a c Ext- a i c ε i i i Ext- c a c Ext- a j c ε j j j Ext- a + ε Ext- a ε Fo Equatio (237 by Theoe 2 we obtai iecty (2 Fo Equatio (232 we obtai Ext- c a c Ext- a c ε Ext- c a c Ext- c a c ε Ext- a + ε i i Ext- a ε j j Fo Equatio (239 by Theoe 2 we obtai iecty Ext- c a c Ext- a i c ε i i i Ext- c a c Ext- a j c ε i j i a (235 (236 (237 (238 (239 (24 Rea 23 Note that we have ove Equatio (235 a Equatio (236 without ay efeece to the 64

19 J Fouzo Lea 24 Defiitio 28 (i Let { α} be a coutabe sequece : such that ( [ α ] a ( ( α ( a a (24 The extea coutabe ue su (-su of the coutabe sequece : is efie by I aticua if { α } { a} the coutabe sequece α : (ii Let { } α α Ext- α α + Ext- α Ext- α su α whee a is efie by α + Ext- a Ext- a Ext-α su a be a coutabe sequece : such that α a (242 the extea coutabe ue su (-su of ( [ α ] a ( ( α ( < a a The extea coutabe owe su (-su of the coutabe sequece a : I aticua if { α } { a} the coutabe sequece α : Ext- α α + Ext- α Ext- α if α whee a Theoe 27 (i Let { α} is efie by The fo ay c the equaity is satisfie (ii Let { α} + (243 (244 is efie by (245 the extea coutabe owe su (-su of α + Ext- a Ext- a Ext-α if a (246 be a coutabe sequece : such that vai the oety (24 α c Ext- α Ext c α c a + Ext- c a be a coutabe sequece : such that vai the oety (244 α The fo ay c the equaity is satisfie + c Ext- α Ext- c α c a + Ext- c a Poof Ieiatey fo Defiitio 28 by Theoe 2 Defiitio 29 Let { z } { a b } (247 (248 + be a coutabe sequece z a + ib : such that ifiite 65

20 J Fouzo seies z absoutey coveges i The extea coutabe coex su (-su of the coesoig coutabe sequece z : is efie by Ext- z Ext- a + i Ext- b Ext- z Ext- a + i Ext- b Ext- z Ext- a + i Ext- b coesoigy Note that ay oeties of this su ieiatey foow fo the oeties of the ea extea su Defiitio 22 (i We efie ow Wattebeg coex ae by i with (249 2 i Thus fo ay z we obtai z x + iy whee xy (ii fo ay z such that z x + iy we efie 2 z by z x + y Theoe 28 Let { z } { a ib } seies (i (ii + be a coutabe sequece z a + ib : such that ifiite z absoutey coveges i to z ζ+ iζ2 a z The ( ζ ( ζ2 ε ( ( ζ ( ζ2 ε ( ( ( 2 ζ ε ζ ε + + Ext- z Ext- a i Ext- b i + i + i Ext- z Ext- a + i Ext-b + i + + i Ext- z Ext a + i Ext- b ( ζ i( ζ2 ε ( i ( ζ + ( ζ2 ε ( + Ext- z Ext- a i Ext- b i i ( ζ + ( ζ2 + ε ( + Ext- z Ext- a i Ext b i i 2 2 ( ζ ( ζ2 Ext- z Ext- a + i Ext- b + i + ε + i 29 Gosho Tasfe Defiitio 22 [7] Let [ S] { x y( y S[ x y] } Note that [ ] (ii S has o axiu the [ ] ( 2 S satisfies the usua axios fo a cosue oeatoie if (i S S a S Let f be a cotiuous sticty iceasig fuctio i each vaiabe fo a subset of ito Secificay we wat the oai to be the catesia ouct A xx a fo soe a By Robi- so tasfe f extes to a fuctio f : whee i { i} i A i fo the coesoig subset of i ito which is aso sticty iceasig i each vaiabe a cotiuous i the Q tooogy (ie ε a δ age ove abitay ositive eeets i We ow exte f to f 66

21 J Fouzo Defiitio 222 [7] Let α a the α i f : (25 i i ( ( b i { } f α α α f b b b a < b α ( i i i Theoe 29 [7] If f a g ae fuctios of oe vaiabe the ( ( ( ( ( f g α f α g ( α Theoe 22 [7] Let f be a fuctio of two vaiabes The fo ay α a a ( α ( f a f bc b α c a < (252 (253 Theoe 22 [7] Let f a g be ay two tes obtaie by coositios of sticty iceasig cotiuous fuctios ossiby cotaiig aaetes i The ay eatio f g o f < g vai i extes to ie Rea 24 Fo ay fuctio f : ( α ( α ( α < ( α f g o f g Theoe 222 [7] ( Fo ay ab Fo ay α β α β (2 Fo ay ab (3 Fo ay αβγ αβγ (4 Fo ay a + we ofte wite fo shot ( a b ( a ( b ( ex ( a b ex ( ba ex + ex ex ex ( α β ex ( α ex ( β ( ex ( α β ex ( βα + b b ( ( a a β ( α γ f istea of f (254 (255 (256 (257 ( ( a ( ( a γβ α (258 ex a a ex (259 Note that we ust aways bewae of the estictio i the oai whe it coes to utiicatio α ex α as the set of aitive ieotets oto the set of a Theoe 223 [7] The a [ ] ( utiicative ieotets othe tha 3 The Poof of the -Tasceee of the Nubes e I this sectio we wi ove the -tasceece of the ubes e Key iea of this oof euctio of the stateet of e is -tasceeta ube to equivaet stateet i by usig seuoig of Wattebeg hyeeas [6] a Gosho ieotet theoy [7] We obtai this euctio by thee stes see Subsectios

22 J Fouzo 3 The Basic Defiitios of the Shiovsy Quatities a I this sectio we ei the basic efiitios of the Shiovsy quatities [8] Let ( ( ε ( be the Shiovsy quatities: + x x ( x ( x e ( x (! ( ( (! (3 + x x x x e ( e x 2 (32 ( ( (! x x x x e ε ( e x 2 (33 whee this is ay ie ube Usig Equatios (3-(33 by sie cacuatio oe obtais: a cosequety ( ε ( ( + e 2 (34 ( + ε ( ( Lea 3 [8] Let be a ie ube The ( ( ( Poof ([8] 28 By sie cacuatio oe obtais the equaity e 2 (35! + Θ Θ ( + µ ( ( ( ( + µ µ + x x x! x c x ( cµ µ + + µ x whee is a ie By usig equaity Γ ( µ x e x ( µ (36 oe obtais Thus (36! whee µ fo Equatios (3 a Γ( ( ( + ( ( (! + µ µ + ( (! c c ( ( ( + Γ( µ ( c!! ! + Θ Θ ( ( ( ( ( (37! + Θ Θ (38 2 Θ2 2 + fo Equatio (33 oe obtais Lea 32 [8] Let be a ie ube The ( Θ ( ( Poof ([8] 28 By subsitutio x u x u By usig equaity ( + ( + ( + (! + u u u u u e ( u 2 (39 ( + µ µ µ + ( ( ( u+ u+ u u+ u ( µ µ + + a by subsitutio Equatio (3 ito RHS of the Equatio (39 oe obtais (3 68

23 J Fouzo ( ( ( 2 + ( + µ µ u u Θ2(! µ + Θ 2 Lea 33 [8] (i Thee exists sequeces a( a g( such that (3 ε ( ( a( (! g whee sequeces a( a g( oes ot ee o ube (ii Fo ay ε ( if (32 : Poof ([8] 29 Obviousy thee exists sequeces a( a g( such that a( a g( oes ot ee o ube a ( ( ( x x x < a x (33 x+ ( ( ( x x e < g x 2 (34 Substitutio iequaities (33-(34 ito RHS of the Equatio (33 by sie cacuatio gives ( ( ( ( ( a g a ε ( g( x! (35! Stateet (i foows fo (35 Stateet (ii ieiatey foows fo a stateet (ii Lea 34 [8] Fo ay a fo ay δ such that < δ < thee exists such that Poof Fo Equatio (35 oe obtais e ( ( e < δ (36 ( ( ( ( ε (37 Fo Equatio (37 by usig Lea 33 (ii oe obtais (37 Rea 3 We ei ow the oof of the tasceece of e foowig Shiovsy oof is give i his boo [8] Theoe 3 The ube e is tasceeta Poof ([8] Suose ow that e is a agebaic ube; the it satisfies soe eatio of the fo a + ae (38 whee a a a iteges a whee a Havig substitute RHS of the Equatio (35 ito Equatio (38 oe obtais ( ( ( + ε ε ( ( ( (39 ( a + a a + a + a Fo Equatio (39 oe obtais ( + ( + ε ( (32 a a a 69

24 J Fouzo We ewite the Equatio (32 fo shot i the fo ( + ( + ε ( a a a ( ( ε ( ( ( a +Ξ + a Ξ a We choose ow the iteges ( ( ( 2 a ( Note that ( a theefoe such that: ( ( ( 2 (32 whee a (322 Ξ Thus oe obtais ( +Ξ( a (323 ( ( ( ( a +Ξ whee a +Ξ (324 By usig Lea 34 fo ay δ such that δ Fo (325 a Equatio (32 we obtai < < we ca choose a ie ube ( δ such that: aε ( < δ a < (325 ( ( a +Ξ + (326 Fo (326 a Equatio (324 oe obtais the cotaictiothis cotaictio fiaize the oof 32 The Poof of the -Tasceee of the Nubes ito Fou Pats e We Wi Divie the Poof 32 Pat I The Robiso Tasfe of the Shiovsy Quatities ( ( ε ( I this subsectio we wi eace usig Robiso tasfe the Shiovsy quatities ( ( ε ( by coesoig ostaa quatities ( ( ε ( The oeties of the ostaa quatities ( ( ε ( eties of the staa quatities ( ( ε ( usig Robiso tasfe icie [4] [5] oe obtais iecty fo the o- Usig Robiso tasfe icie [4] [5] fo Equatio (38 oe obtais iecty ( ( ( ( (! + Θ Θ \ (327 Fo Equatio (3 usig Robiso tasfe icie oe obtais ( : ( ( 2( 2( Θ Θ 2 (328 Usig Robiso tasfe icie fo iequaity (35 oe obtais ( : ( ( g( a( ε ( (! Usig Robiso tasfe icie fo Equatio (35 oe obtais ( : ( + ε ( Lea 35 Let 2 ( ( ( e e 2 ( the fo ay a fo ay δ δ thee exists ( ( e < δ (329 (33 such that (33 6

25 J Fouzo Poof Fo Equatio (33 we obtai ( : ( ( Fo Equatio (332 a (329 we obtai ( Pat II The Wattebeg Ibeig ( e ( ( e ε (332 ito I this subsectio we wi eace by usig Wattebeg ibeig [6] a Gosho tasfe the ostaa e a the ostaa Shiovsy quatities ( ( ( by coeso- quatities ( ig Wattebeg quatities ( e ( ( ( ( ( ε ( quatities ( ( ( ( ( e ( ( ε esoig ostaa quatities ( e ( ( ε ( [7] By usig Wattebeg ibeig The oeties of the Wattebeg oe obtais iecty fo the oeties of the co- usig Gosho tasfe icie [4] fo Equatio (33 oe obtais ( + ε ( ( e ( e ( 2 ; 2 By usig Wattebeg ibeig fo Equatio (327 oe obtais ε (333 a Gosho tasfe (see Subsectio 29 Theoe 29 ( ( (! + Θ( ( ( ( ( ( + Θ Θ! 3 By usig Wattebeg ibeig ( ( ( ( fo Equatio (328 oe obtais Θ2 Θ2 (334 2 (335 Lea 36 Let the fo ay a fo ay δ δ thee exists ( such that ( e < δ (336 ( Poof Iequaity (336 ieiatey foows fo iequaity (33 by usig Wattebeg ibeig a Gosho tasfe 323 Pat III Reuctio of the Stateet of e Is -Tasceeta Nube to Equivaet Stateet i Usig Gosho Ieotet Theoy To ove that e is -tasceeta ube we ust show that e is ot w-tasceeta ie thee oes ot exist ea -aaytic fuctio ( g x ax with atioa coefficiets a a a such that ae a e (337 6

26 J Fouzo Suose that e is w-tasceeta ie thee exists a -aaytic fuctio ( coefficiets: such that the equaity is satisfie: g x ax with atioa a a a a (338 ae (339 a e I this subsectio we obtai a euctio of the equaity give by Equatio (339 to equivaet equaity give by Equatio (3 The ai too of such euctio that extea coutabe su efie i Subsectio 28 be the su coesoigy Lea 37 Let ( a ( The ( 2 ( a + ae ae ( + Poof Suose thee exists such that ( The fo Equatio (339 foows ( foe by Theoe 3 oe obtais the cotaictio that (34 Thee- Rea 32 Note that fo Equatio (339 foows that i geee case thee is a sequece { i} i o thee is a sequece { j} j o both sequeces { i} i such i i i i ( i ae < a + i ae (34 i i such that j j i j ( j ae a + i ae i j (342 a { j} j with a oety that is secifie above exist Rea 33 We assue ow fo shot but without oss of geeeity that (34 is satisfie The fo (34 by usig Defiitio 27 a Theoe 24 (see Subsectio 28 oe obtais the equaity [4] Rea 34 Let ( a ( ( ( ( a + Ext- a e ε (343 be the ue extea su efie by + Ext- a e ( a ( a ( e ( + Note that fo Equatio (343-Equatio (344 foows that ( ( (344 + ε (345 Rea 35 Assue that αβ a β I this subsectio we wi wite fo a shot ab αβ 62

27 J Fouzo iff β absobs α ie β + α β Lea 38 ab ( ( 2 Poof Suose thee exists such that ( ( ab The fo Equatio(345 oe obtais ( ε (346 Fo Equatio (346 by Theoe 2 foows that ( a theefoe by Lea 37 oe obtais the cotaictio Theoe 32 [4] The equaity (343 is icosistet Poof Let us cosie hyeatua ube I efie by coutabe sequece ( I (347 Fo Equatio (343 a Equatio (347 oe obtais ( ( ( I a +I Ext- a e I ε (348 Rea 36 Note that fo iequaity (327 by Wattebeg tasfe oe obtais g ( a( ε ( (! Substitutio Equatio (33 ito Equatio (348 gives ( ( ε Ext- ( ( e Ext- + I + I I + ( I I ε \ { } ( ( a ( a I I I I utiyig Equatio (35 by Wattebeg hyeitege ( tio 28 oe obtais ( (349 (35 by Theoe 23 (see subsec- { } ( Ext- ( ( ε ( I + I +I I ε By usig iequaity (349 fo a give δ δ we wi choose ifiite ie itege that: ( ( ( (35 such Ext- I ε I δ ε (352 Now usig the iequaity (349 we ae fee to choose a ie hyeitege ( δ δ i the Equatio (35 fo a give such that: ( δ ( Hece fo Equatio (352 a Equatio (353 we obtai a δ I (353 ( ( Ext- I ε ε (354 Theefoe fo Equatios (35 a (354 by usig efiitio (25 of the fuctio ( α It give by Equatio (22-Equatio (22 a coesoig basic oety I (see Subsectio 27 of the fuctio 63

28 J Fouzo It ( α we obtai ( Ext { ( ε ( } ( Ext- { ( } ( ( ε ( ε It I + - I +I I + I It I I Fo Equatio (355 usig basic oety I of the fuctio It ( α fiay we obtai the ai equaity { } ( ( ( ( (355 I + Ext- I I ε (356 We wi choose ow ifiite ie itege i Equatio (356 ˆ such that Hece fo Equatio (334 foows Note that ( ˆ Usig Equatio(335 oe obtais ( ˆ ax I (357 ˆ ( ˆ (358 Usig (357 a (358 oe obtais: ˆ ( ˆ ( I ˆ (359 ( ˆ 2 ( Pat IV The Poof of the Icosistecy of the ai Equaity (356 I this subsectio we wi ove that ai equaity (356 is icosistet This ooff base o the Theoe 2 (v see Subsectio 26 Lea 39 The equaity (356 ue coitios (359-(36 is icosistet Poof (I Let us ewite Equatio (356 i the shot fo Γ ( ˆ + Σ ( ˆ Λ ( ˆ ε (36 whee Fo (359-(36 foows that ( ˆ ( ˆ Γ I ( ˆ ( ˆ { } Σ ( ˆ Ext- ( I ( ˆ Λ I ˆ ˆ Γ Σ ( ˆ ( ˆ ( Γ +Σ { } ( ˆ Γ + Σ ˆ Λ ( ˆ ε Rea 37 Note that Σ ( ˆ Othewise we obtai that ( ˆ ( ˆ othe ha fo Equatio (36 foows that ( ( (362 (363 ab But the ab But this is a cotaic- tio This cotaictio coete the oof of the stateet (I ˆ ˆ ˆ ˆ ε ˆ ε be the extea su (II Let ( ( ( 2 a ( ( 64

29 J Fouzo coesoigy a { } { ˆ } ( ˆ ( ˆ ( ˆ Γ + I ( ˆ Ext- I ( { ( ˆ I } ( ˆ ε ( ˆ { ( ˆ Γ + I +I ε( } ˆ ( ε - { ( ˆ Ext I +I ε( } + Note that fo Equatio (36 a Equatio (364 foows that Lea 3 Ue coitios (359-(36 ( ˆ ( ˆ ( ˆ + Λ ε (364 (365 ( ˆ ε ( ˆ ε ab 2 (366 ( ˆ ( ˆ Poof Fist ote that ue coitios (359-(36 oe obtais Suose that thee exists a oe obtais Fo Equatio (369 by Theoe 27 oe obtais Thus ab 2 (367 ( ˆ ε (368 The fo Equatio (365 such that ab ( ˆ ( ˆ ε ε ( ( ˆ ˆ (369 ε Λ ε ( ˆ ( ˆ ( ˆ ( ˆ ( ˆ ε ε Λ Λ ε (37 ( ε (37 ˆ Fo Equatio (37 by Theoe 2 foows that ( cotaictio This cotaictio fiaize the oof of the Lea 3 Pat (III a theefoe by Lea 37 oe obtais the Rea 38 (i Note that fo Equatio (362 by Theoe 2 (v fows that Σ ( ˆ whee ( ( ( ( ( ε has the fo ˆ q ab ˆ q ˆ (372 Σ + Σ + Λ ( ( q Σ ˆ ˆ q a q ˆ (373 (ii Substitutio by Equatio (372 ito Equatio (36 gives ( ( ( ( ( ε ( ˆ ˆ ˆ q ˆ ˆ ε (374 Γ + Σ Γ + + Λ Λ 65

30 J Fouzo Rea 39 Note that fo (374 by efiitios foows that ( ( ˆ ( ˆ ( ε ab Γ + q Λ (375 Rea 3 Note that fo (373 by costuctio of the Wattebeg itege Σ ( ˆ that thee exists soe such that Theefoe ( ˆ q ( ˆ obviousy foows < < (376 ( ˆ ( ˆ ( ˆ q ( ˆ ( ˆ Γ + <Γ + Γ + (377 Note that ue coitios (359-(36 a (373 obviousy oe obtais ( ˆ ( ˆ ( ˆ ( ˆ ( ˆ ( ˆ (378 Γ + <Γ + q Γ + Γ + q Fo Equatio(374 foows that Theefoe Fo (378 foows that ( ( ( ε ( ˆ q ˆ ˆ ε (379 Γ + + Λ Λ ( ( ˆ ( ˆ ( ε Λ Γ + q + ε (38 ( ( ( ( ( ( Λ ( ˆ Γ ( ˆ + ( ˆ ( Λ ( ˆ Γ ( ˆ + q ( ( Λ ˆ Γ ˆ + ˆ < Λ ˆ Γ ˆ + q Note that by Theoe 28 (see Subsectio 25 a Foua (344 oe otais {( ( ( ( } ˆ ˆ ˆ ( ( ˆ a ( ( ˆ ( ˆ Λ Γ + ( ˆ a ˆ + Wst Λ Γ + + Wst { } ( ( {( ( ( } Λ ˆ Γ ˆ + (38 Wst Wst (382 Wst q Fo Equatio (38-Equatio (382 foows that Thus a theefoe { } ( a ( ˆ ( ( ˆ ( ˆ Wst + < Wst Λ Γ + q ( a ( ˆ Wst + {( ( ( } Λ ˆ Γ ˆ + Wst q ( ( ˆ ( ˆ ( ε (383 ab Λ Γ + q (384 ( ( ˆ ( ˆ ( ε Λ Γ + q + ε (385 But this is a cotaictio This cotaictio coete the oof of the Lea 39 66

31 J Fouzo 4 Geeaize Shiovsy Quatities I this sectio we ei the basic efiitios of the Shiovsy quatities see [8] Theoe 4 [8] Let f ( z 2 be a oyoias with coefficiets i Assue that fo ay 2 agebaic ubes ove the fie : β β 2 fo a coete set of the f z such that oots of ( a a 2 a The: Let f ( z be a oyoia such that ( [ ] ( f z z eg f z 2 (4 + a a e β (42 N ( ( ( β f z f z b + bz + + b z b z b b N (43 N N N Let ( N ( N a ( N ε be the quatities [8]: ( (! ( N z N + b z f z e z ( N (44 whee i (44 we itegate i coex ae aog ie [ + ] see Pictue ( ( (! ( N z N b z f z e z (45 + β β N e whee a whee i (45 we itegate i coex ae aog ie with iitia oit β a which ae aae to ea axis of the coex ae see Pictue ε ( ( (! ( N z N β β b z f z e z N e (46 whee a whee i (46 we itegate i coex ae aog cotou β Fo Equatio (43 oe obtais whee ( ( N + ( N ( N s N ( N s s + see Pictue b z f z b b z + c z (47 bn b cs s N Now fo Equatio (44 a Equatio (47 usig foua s ( x ( Γ s x e x s! s Pictue Cotou β i coex ae 67

32 J Fouzo oe obtais whee b b C N foows that ( N + ( N + + N z s s z b b c ( N z e z+ z e z!! ( ( N + ( s s + ( ( s + ( N! ( N bn b + c s bn b + C! We choose ow a ie such that ( a b b Fo Equatio (43 a Equatio (45 oe obtais ( (48 ax The fo Equatio (48 a N (49 β + j e N z+ β ( N bn z z ( z βi j e z (! (4 β j i whee By chage of the vaiabe itegatio z u+ β i RHS of the Equatio (4 we obtai + j N u ( N bn ( u β ( u e z β β + + i j u (! (4 j i j i whee Let us ewite ow Equatio (4 i the foowig fo Let + j u ( N ( bn u b ( N β u e bn u b N β bn β + + i j u (! (42 j i j i A be a ig of the a agebaic iteges Note that [8] αi j bnβi j A i j j (43 Let us ewite ow Equatio (42 i the foowig fo ( + j u ( N ( b u+ α u e ( b u+ α α u (44 N N i j! j i j i whee Fo Equatio (44 oe obtais The oioia ( u u Φ ( u (! u e a ( N u Φ + + j u ( u a ( b u α u e ( b u α α N N i j j i j i Φ is a syetic oyoia o ay syste { α α2 α } α α α 2 It we ow that Φ ( u [ u] [8] a theefoe ( N + s ( s s s + A N (45 of vaiabes α α2 α whee (46 u Φ u c u c (47 68

33 J Fouzo Fo Equatio (45 a Equatio (47 oe obtais Theefoe Let Φ ( ( ( N + ( s s s + ( u ( N + u e u cs s u (! s + (! a N u u e u! c C C! Ξ Ξ (48 ( N a ( N ( N (49 OR be a cice wth the cete at oit ( We assue ow that ( β OR esigate ow ( ( z+β ax N zr g b f z e Fo Equatio (46 a Equatio (42 oe obtais ε ( ( ( ( g ax g g ax b zf z ( N whee Note that zr ( (! ( N N β z+ β b z f z e z (! ( ( ( ( ( (! Fo (422 foows that fo ay [ δ ] N ( ( ( We wi (42 β z β + bn f ( z e bn zf ( z z (42 g g β g g R!! g g R if thee exists a ie ube such that (422 aε ( N ( < (423 whee Fo Equatio (44-Equatio (46 foows whee Assue ow that e ( + ε ( ( N N N (424 β + a a e β (425 Havig substitute RHS of the Equatio (424 ito Equatio (425 oe obtais ( ( ( N + ε N N ε ( N (426 ( N ( N ( N a + a a + a + a Fo Equatio (426 by usig Equatio (49 oe obtais ( ε ( (427 a +Ξ N + a N 69

34 J Fouzo We choose ow a ie such that ax ( a b b N a ( < Note that Ξ ( N a theefoe fo Equatio (49 a Equatio (427 oe obtais the cotaictio This cotaictio coete the oof 5 Geeaize Liea-Weiestass Theoe Theoe 5 [4] Let f ( z 2 be a oyoias with coefficiets i Assue that fo ay agebaic ubes ove the fie : β β 2 fo a coete set of the oots of f ( z such that ( [ ] ( f z z eg f z 2 (5 a a a 2 2 We assue ow that The a e β < (52 + a a e β (53 We wi ivie the oof ito thee ats Pat I The Robiso tasfe Let f ( z f ( z [ z] z 2 be a ostaa oyoia such that ( ( ( N ( f z f z f z b + b z+ + b z ( z ( β b b b N N Let ( N ( N a ( ( N ε N be the quatities: ( (! ( + ( N z N N N N (54 b z f z e z N (55 whee i (55 we itegate i ostaa coex ae aog ie [ ] ( N ( ( (! + see Pictue ( + ( N z b z f z e z N N (56 e β β whee a whee i (56 we itegate i ostaa coex ae aog ie with iitia oit β a which ae aae to ea axis of the coex ae see Pictue β ( N z b z f z e z N ε N (! β ( N ( e ( (57 whee a whee i (57 we itegate i ostaa coex ae aog cotou β see Pictue Usig Robiso tasfe icie [4]-[6] fo Equatio (55 a Equatio (48 oe obtais iecty ( ( N b N N b + C (58 62

35 J Fouzo whee b b C We choose ow ifiite ie N { N ( such that ax a b b (59 2 Usig Robiso tasfe icie fo Equatio (56 a Equatio (49 oe obtais iecty a theefoe ( ( ( ( ( : Ξ N a N C (5 ( ( N : Ξ (5 3 Usig Robiso tasfe icie fo Equatio (57 a Equatio (42 oe obtais iecty ε whee β ( N ( e β β b (! N ( ( ( ( ( (! ( N z N z f z e z ( β z + b f z e bn zf ( z z ( ( ( g g β g g!! Note that ( [ ] thee exists ( 4 Fo (53 foows that fo ay [ δ ] whee ( g ( (! (52 g (53 thee exists a ifiite ie ( ( ( ( ( N 5 Fo Equatio (55-Equatio (57 we obtai whee such that : a ε < (54 Pat II The Wattebeg ibeig By usig Wattebeg ibeig fo Equatio (58 oe obtais whee N e ( ( N + ( N ( N ε β ( ( ( N (55 e β ito a Gosho tasfe (see Subsectio 28 Theoe 27 ( ( ( N ( N N + + N b b C b b C (56 b b C We choose ow a ifiite ie 2 By usig Wattebeg ibeig iecty ( ( { N ( such that ax a b b (57 ( ( a Gosho tasfe fo Equatio (5 oe obtais ( ( ( N N C : Ξ a (58 62

36 J Fouzo a theefoe 3 By usig Wattebeg ibeig iecty ( ( Ξ( N (59 ( ( 4 By usig Wattebeg ibeig iecty whee Pat III ai equaity a Gosho tasfe fo Equatio (54 oe obtais ( ( ( ( N : a ε ε < (52 e ( e a Gosho tasfe fo Equatio (55 oe obtais ( ( N + ( N ( N ( ( ε β β (52 Rea 5 Note that i this subsectio we ofte wite fo a shot we wite ( : e ( N + ε ( N ( N β a istea ( a a Fo exae istea Equatio (52 Assutio 5 Let f ( z 2 be a oyoias with coefficiets i Assue that fo ay agebaic ubes ove the fie : β β 2 fo a coete set of the oots f z such that of ( ( [ ] ( f z z eg f z 2 (522 2 a a 2 Note that fo Assutio 5 foows that agebaic ubes ove the fie : ( ( β β β β 2 fo ay 2 fo a coete set of the oots of ( [ ] ( ( ( f z f z z eg f z 2 (523 Assutio 52 We assue ow that thee exists a sequece a atioa ube such that a q a 2 (524 a q a e β (525 < (526 a a e β + (

37 J Fouzo Assutio 53 We assue ow fo a shot but without oss of geeaity that the a ubes β β 2 ae ea I this subsectio we obtai a euctio of the equaity give by Equatio (527 i to soe equivaet equaity give by Equatio (3 i The ai too of such euctio that extea coutabe su efie i Subsectio 28 be the su coesoigy Lea 5 Let ( a ( ( a + a e β ( a e + The ( 2 Poof Suose thee exists such that ( The fo Equatio (527 foows ( foe by Theoe 4 oe obtais the cotaictio that β (528 Thee- Rea 52 Note that fo Equatio (527 foows that i geee case thee is a sequece { i } i i β i i ( i a + ae < i a + a e i β i i such (529 o thee is a sequece { j} j such that j β i j ( j a + ae i a + a e j β i j (53 o both sequeces { i} i a { j} j with a oety that is secifie above exist Rea 53 We assue ow fo shot but without oss of geeeity that (529 is satisfie The fo (529 by usig Defiitio 27 a Theoe 24 (see Subsectio 28 oe obtais the equaity [4] Rea 54 Let ( a ( ( ( ( a Ext- a e β + ε be the ue extea su efie by + - β ( a ( a ( e β ( Ext ( a ( e + Note that fo Equatio (53-Equatio (532 foows that ( ( (53 (532 + ε (533 Rea 55 Assue that α β a β I this subsectio we wi wite fo a shot iff β absobs α ie β + α β Lea 52 ab ( ( 2 ab αβ 623

38 J Fouzo Poof Suose thee exists such that ab ( ( The fo Equatio (533 oe obtais ( (534 ε Fo Equatio (534 by Theoe 2 foows that ( a theefoe by Lea 5 oe obtais the cotaictio Theoe 52 [4] The equaity (53 is icosistet Poof Let us cosiee hyeatua ube I efie by coutabe sequece Fo Equatio (53 a Equatio (535 oe obtais whee ( I (535 I +I I + I I ε β β ( a Ext- ( a ( e Ext- ( e I q I I a I q I I a Rea 56 Note that fo iequaity (52 by Gosho tasfe oe obtais ( ( g g β ε ( N Substitutio Equatio (52 ito Equatio (536 gives ( (536 (537 N (538! ( N + ε ( N I + Ext I I ε ( N utiyig Equatio (539 by Wattebeg hyeitege ( N sectio 28 we obtai - N Ext ( - ( N ε ( N ( N I (539 by Theoe 23 (see sub- I + I + (54 ε By usig iequaity (538 fo a give δ δ we wi choose ifiite ie itege δ such that: ( I ( N Ext- ε δ ε Theefoe fo Equatios (54 a (54 by usig efiitio (25 of the fuctio ( α (54 It give by Equatio (22-Equatio (22 a coesoig basic oety I (see Subsectio 27 of the fuctio It α we obtai ( It I ( N + Ext- I ( N + ε ( N Ext ( N - ( N ( ( ε ( I + I It I N I N ε Fo Equatio (542 fiay we obtai the ai equaity (

39 J Fouzo ( N ( ( N N I + Ext- I I ε We wi choose ow ifiite ie itege i Equatio (356 ˆ such that Hece fo Equatio (56 foows Note that ( ˆ Usig Equatio (5 oe obtais ( N ˆ ax I (543 a b b (544 ˆ ( Nˆ Usig (544 a (545 oe obtais: (545 ( ˆ ˆ N I (546 ( ˆ ˆ N 2 (547 Pat IV The oof of the icosistecy of the ai equaity (543 I this subsectio we wi ove that ai equaity (543 is icosistet This oof is base o the Theoe 2 (v see Subsectio 26 Lea 53 The equaity (543 ue coitios (546-(547 is icosistet Poof (I Let us ewite Equatio (543 i the shot fo whee ( ˆ ( ˆ ( ˆ Γ N + Σ N Λ ε (548 Σ ( N ˆ Ext- I ( N (549 Γ ( ˆ I ( ˆ Λ ( ˆ I ( ˆ N N Fo (546-(547 foows that Rea 57 Note that Σ ( Nˆ ˆ ˆ Γ Σ ( Nˆ ( Nˆ Othewise we obtai that But the othe ha fo Equatio (548 foows that ( Γ ( ˆ +Σ ( ˆ { } (55 ab N N (55 ( ( ˆ ( ˆ ( ˆ ab Γ N + Σ N Λ ε (552 But this is a cotaictio This cotaictio coete the oof of the stateet (I N ˆ N ˆ N ˆ N ˆ ε N ˆ ε be the extea su (II Let ( ( ( 2 a ( ( coesoigy ( ˆ ( ˆ N Γ N + I ( N ( N ˆ I ( N + 2 ( ˆ 2 N I ( N Γ + I + I + + ( N ˆ ε ( ˆ N { ( N ε N } ( N ˆ ε Ext- { ( N ε( N } ( (

40 J Fouzo a Note that fo Equatio (543 a Equatio (553 foows that ( Nˆ ( Nˆ ( ˆ + Λ ε 2 (554 Lea 54 Ue coitios (546-(547 ( ˆ ε ( ˆ ε ab N N 2 (555 ( ˆ ( ˆ Poof Fist ote that ue coitios (546-(547 oe obtais Suose that thee exists oe obtais ab N N 2 (556 ( Nˆ ε 2 (557 ab ˆ ˆ Nε N ε he fo Equatio (554 such that ( ( ( Nˆ ( ˆ (558 ε Λ ε Fo Equatio (558 by Theoe 27 oe obtais Thus ( ˆ ( ˆ ( ˆ ( ˆ ( ˆ ε ε Λ N Λ N ε N ε (559 ε ( ˆ ε Fo Equatio (56 by Theoe 2 foows that ( cotaictio This cotaictio fiaize the oof of the Lea 54 (III N (56 a theefoe by Lea 52 oe obtais the Rea 58 (i Note that fo Equatio (549 by Theoe 2 (v fows that Σ ( ˆ whee ( ( ( ( ( N has the fo Σ N ˆ q + ab Σ N ˆ q + Λ ˆ ε (56 ( ( q Σ N ˆ N ˆ q a q ˆ (562 (ii Substitutio by Equatio (56 ito Equatio (548 gives ( ( ( ( ( ε ( N ˆ N ˆ N ˆ q ˆ ˆ ε (563 Γ + Σ Γ + + Λ Λ Rea 59 Note that fo (563 by efiitios foows that ( ( ˆ ( ˆ ( ε ab Γ N + q Λ (564 Rea 5 Note that fo (562 by costuctio of the Wattebeg itege Σ ( ˆ that thee exists soe 2 such that Theefoe ( Nˆ q ( Nˆ 2 2 N obviousy foows < < (565 ( Nˆ ( Nˆ ( Nˆ q ( Nˆ ( Nˆ Γ + <Γ + Γ + (566 2 Note that ue coitios (546-(547 a (566 obviousy oe obtais ( ˆ ( ˆ ( ˆ ( ˆ ( ˆ ( ˆ Γ N + N <Γ N + q Γ N + N Γ N + q (567 Fo Equatio (563 foows that 2 626

41 J Fouzo Theefoe Fo (569 foows that ( ( ( ε ( N ˆ q ˆ ˆ ε (568 Γ + + Λ Λ ( ( ˆ ( ˆ ( ε Λ Γ N + q + ε (569 ( ( ( ( ( ( Λ ( ˆ Γ ( N ˆ + ( ˆ 2 N ( Λ ( ˆ Γ ( N ˆ + q ( ( Λ ˆ Γ N ˆ + N ˆ < Λ ˆ Γ N ˆ + q Note that fo (57 by Theoe 28 (see Subsectio 25 a Foua (532 oe otais {( ( ( ( } ˆ ˆ ˆ ( ( ˆ a ( ( ˆ ( ˆ Λ Γ + ( ˆ a ˆ Wst Λ Γ + + N N Wst { } ( ( {( ( ( } Λ ˆ Γ ˆ + (57 Wst N Wst N (57 Wst q Fo Equatio (57-Equatio (57 foows that Wst ( a + ( ˆ ( ( ˆ ( ˆ N < Wst Λ Γ + q Wst ( a + ( ˆ 2 N Thus a theefoe {( ( ( } Λ ˆ Γ ˆ + Wst N q { } ( ( ˆ ( ˆ ( ε (572 ab Λ Γ + N q (573 ( ( ˆ ( ˆ ( ε Λ Γ N + q + ε (574 But this is a cotaictio This cotaictio coete the oof of the Lea 53 Rea 5 Note that by Defiitio 28 a Theoe 28 fo Assutio 5 a Assutio 52 foows 2 2 ( - ( ( a + Ext a e β ε ε Theoe 53The equaity (575 is icosistet Poof The oof of the Theoe 53 obviousy coies i ai etais the oof of the Theoe 53 Theoe 53 coete the oof of the ai Theoe 6 Refeeces [] Nesteeo YV a Phiio P Es (2 Itouctio to Agebaic Ieeece Theoy Seies: Lectue Notes i atheatics Vo 752 XIII Sige Sciece & Busiess eia 256 [2] Waschit (23 Agebaic Vaues of Aaytic Fuctios Joua of Coutatioa a Aie atheatics htt://xoiog/6/s (3637-x [3] Fouzo J (26 The Soutio of oe Vey O Pobe i Tasceeta Nubes Theoy Sig Ceta Sec- 627

42 J Fouzo tioa eetig Note Dae IN 8-9 Ai 26 eetig 6 Peiiay Reot htt://wwwasog/eetigs/sectioa/6--8f [4] Fouzo J (23 No-Achieea Aaysis o the Extee Hyeea Lie a Soe Tasceece Cojectues ove Fie a ω htt://axivog/abs/97467 [5] Gobatt R (998 Lectues o the Hyeeas Sige-Veag New Yo htt://xoiog/7/ [6] Wattebeg F (98 [ ] -Vaue Tasatio Ivaiat easues o a the Deei Coetio of Pacific Joua of atheatics htt://xoiog/24/j [7] Gosho H (985 Reas o the Deei Coetio of a Nostaa oe of the Reas Pacific Joua of atheatics htt://xoiog/24/j98587 [8] Shiovsy AB (982 Diohatie Aoxiatios a Tasceeta Nubes oscow State Uivesity oscov (I Russia htt://eboofiog/boo/5657 htt://booeog/eae?fie5657&g29 628