196.TheTheoryofHardy’sZ-Function.ByA. Ivi ´ c 197.InducedRepresentationsofLocallyCompactGroups.ByE.KANIUTHandK.F. Taylor 198.TopicsinCriticalPointTheory.ByK.PERERAandM. Schechter 199.CombinatoricsofMinusculeRepresentations.ByR.M. Green 200.SingularitiesoftheMinimalModelProgram.ByJ. Koll ´ ar 201.CoherenceinThree-DimensionalCategoryTheory.ByN. Gurski 202.CanonicalRamseyTheoryonPolishSpaces.ByV. Kanovei,M. Sabok,andJ. Zapletal 203.APrimerontheDirichletSpace.ByO.EL-Fallah,K. Kellay,J. Mashreghi,and T. Ransford
InChapter4weprovethatcertaindiophantineequationsintwovariables haveatmostfinitelymanysolutions,usingtheauxiliarypolynomialpioneered byRunge.Themethodenablesallsolutionstobefoundinprinciple.Atypical exampleisthatthereareatmostfinitelymanyintegers x, y with x(x 3 2y 3 ) = y.
Or,comingfromCassels’swell-knownresultontheCatalanEquationrecently solvedcompletelybyMih ˘ ailescu,thereareatmostfinitelymanyintegers x, y with x5 y7 = 1provided y isnotdivisibleby5(wedonotproveMih ˘ ailescu’s Theoremhere).Ofcourseequationslike
x 3 2y 3 = m
forfixed m aremorenatural,andthesewillbeconsideredinChapter12.For theproofshere,weneedtoknowthatthelargecomplexsolutionsaregiven byPuiseux(orbetterLaurent)series.Itseemsthatthisisnotsoeasytofind intheliterature,especiallyregardingthecrucialconvergenceproperties,sowe providequiteafewdetails.
TheninChapter5weprovesomeresultssimilartotheclassicalHilbert IrreducibilityTheorem,usuallyabbreviatedtoHIT,byusingthemachineryof theprecedingchapter.TheyarenotsogeneralasHIT,butwhentheydowork, theydelivermoreinformation.TheresultswerefirstfoundbySprindzhukalso usingauxiliarypolynomials,butinamoreelaborateway.Nowadaysthissort ofthingcanbedonewithheightsmachinery,butthatisnotsoelementary.A typicalexample,relatedtothatofthepreviouschapter,isthatthereareatmost finitelymanyintegers y suchthatthepolynomial
X (X 3 2y 3 ) y
in Q[X ] isreducibleovertherationals,andinprinciplethesecanallbefound. AliteralapplicationofHITwouldshowonlythatthereareinfinitelymany rational y suchthatthepolynomialisnotreducible.Sosometimeswegeta StrongHilbertIrreducibilityTheorem;butwerefrainfromabbreviatingthis. Hereweneedresultants;thesecanbefoundalmostanywhere,butbecausewe usethemfrequentlyinthisbookweprovideaself-containedaccount.
InChapter6wejumptoadifferenttopic.Weprovethatthenumber N of pointsmoduloaprime p onanaffineellipticcurvesatisfies
ThereisafamousresultofP ´ olyaonentirefunctionsmappingthenatural numberstotherationalintegers;thismayhaveinfluencedGelfondinhispioneeringworkonthetranscendenceof α β (seeChapter19).P ´ olya’soriginal proofusedinterpolationformulaeandgavethebestpossibleconstant.Much laterWaldschmidtgaveaversionbyauxiliarypolynomials,whichsadlygives aworseconstant.Theproofisneverthelessilluminating;itneedsbinomial coefficientstoavoidfactorials,oneofthekeyideasinThue’sfamousproof (seeChapter12).Moreprecisely,weshowinChapter9thatanentirefunction f with
allin Z mustbeapolynomialif |f (z)| growsoforderatmost C |z| foracertain C > 1.P ´ olyacouldtakeany C < 2;andthestandardexample2z showsthat nothingbetterispossible.Orreformulated:ifanon-polynomialentirefunction f hasthisgrowth,thenatleastoneof f (0), f (1), f (2), ... mustbenon-integral. Gelfond’sstepfromnon-integralitytotranscendenceneededmanymoreideas, allofwhichwillbedevelopedinthisbook.
TherathernaturalgeneralizationtotheGaussianintegers G = Z + Zi with f mapping G intoitselfalsoplayedasimilarhistoricalrole;forexampleit probablydirectlyinspiredGelfond’sproofofthetranscendenceof eπ .Butthe bestpossibleconstantdidnotappearuntilarelativelyrecentpaperofGramain; paradoxicallyenough,hisproofinvolvesanauxiliarypolynomial(orbetteran auxiliaryfunction).Moreprecisely, f itselfmustbeapolynomialif |f (z)| now growsoforderatmost C|z|2 foracertain C > 1.Gelfondconsideredthis problemtoo,andobtainedthenotoriousvalue
C = exp π
(modestlynotmentionedinhisbook).Inthelate1970s,Iobtainedaconstant, extremelydifficulttocompute,whichlaterturnedouttobeabout1.181;and Iconjecturedthatthebestpossibleconstantwasexp( π 2e ) about1.782.This Gramainproved,andsodoweinChapter10.
Hereismaybeoneofthesimplestexamples. Thereisanoldchestnutwhichoftenturnsupinproblem-solvingsessions: givenapolynomial F inavariable X ,canonealwaysmultiplyitbyanon-zero polynomialtogetaproductinvolvingonlypowers X p for p prime?
Forexamplewith F = X 100 + 1wehave X 3 F = X 103 + X 3 .Butwhatabout F = X 100 + X 3 ?Heremultiplyingbysome P = aX d willnotdo.However (X 111 X 14 )F = X 11 (X 100 X 3 )F = X 11 (X 200 X 6 ) = X 211 X 17 .
Atfirstsightitappearstobeadifficultproblemaboutprimes,possiblyin arithmeticprogressions.Sowhatabout F = X 1000 + X 100 + X 3 ?
Letusconsidermultiplying F bysome
forunknown L andundeterminedcoefficients pi (notnecessarilyprimes,but theymightbe).Then PF hasdegreeatmost L + 1000,anditscoefficientsare linearformsinthe pi .Wewouldliketoeliminatetheterms X n for n notprime with0 ≤ n ≤ L + 1000.Thereare
L + 1001 π(L + 1000) homogeneouslinearequationsinthe L + 1unknowns pi .Bylinearalgebrathissystemissolvablenon-trivially,provided L + 1 > L + 1001 π(L + 1000); thatis, π(L + 1000)> 1000.Everyschoolgirlknowsthatthereareinfinitely manyprimenumbers,so π(x) tendstoinfinitywith x andthereexistssuchan L;forexample L = 6927(withMaple).Sotheanswerisyesforthis F ;the troubleofcourseisthatwehavetosolve6927equationsin6928unknownsto get P explicitly.
Thereadermaynowseefirstthatthisworksforany F ,andsecondthatthe primesareirrelevant,inthesensethatwemaydemandonlypowers X m in PF with m inanyprescribedinfiniteset;forexampletheelementsofthesequence 4,27,3125,823543, ... ofall m = pp .
Howcanweeliminate t ?Commonsense,orageneralconsiderationoftranscendencedegree,showsthattheremustbeanalgebraicrelationbetween x, y notinvolving t .Andindeedamoment’sthoughtgives
0.(1.3)
Butwhatabout
= t 3 + t , y = t 4 + t ?
Wecouldsolvethefirstequationbyradicalsfor t ,andthensubstituteintothe secondequation,andfinallysomehowcleartheradicals.Weget
Thereadermayseefirstthatthisworksforanytwopolynomials F , G in t insteadof(1.5),andsecondthatitgeneralizestomorevariables;forexample betweenanythreepolynomialsintwovariablesthereisanon-trivialalgebraicrelation(aswouldfollowmoresimplybyconsiderationoftranscendence degree).
Thisexampleisperhapsmoretypicalofthosetofollowinthesepages.After thesubstitution(1.5)wemayregardthefunction P(x, y) ashavingalargeorder ofvanishingat t = 0;solarge,indeed,thatitmustvanishidentically.
whosesizeisthenumber n ofunknowns.Suchdeterminantsalreadyhave n! terms,sotheirvaluesarelikelytobesomewhatlarger.Thusitwouldbe surprisingiftheintegersinourrelationweresubstantiallylessthanthefactorial 10491121!.Anditcanbeseenthatsomeoftheentriesofthedeterminantsare almostaslargeas36476 (asin Exercise1.17).Thismeansthatwecouldexpect somecoefficientsin P tohavethirtythousandmillion(Americanthirtybillion, SwissdreissigMilliarden)decimaldigits(seehowever Exercise8.13).Soitis doubtfulif P couldeverbeexpressedexplicitly.
Exercises
1.1 Showthatthereis P = 0in C[X ] suchthat P(X )(X 1000 + X 100 + X 3 ) has theform N n=0 an X n2 .
1.2 Let F bein C[X ] withdegreeatmost D.Showthatthereis P = 0in C[X ] withdegreeatmost D2 D suchthat PF hastheform N n=0 an X n2
1.3 Let F bein C[X ].Find P = 0in C[X ] suchthat PF hastheform N n=0 an X 2n .
1.4 Let F bein C[t ] withdegreeatmost D ≥ 1andlet G bein C[t ] with degreeatmost E ≥ 1.Showthatthereis P = 0in C[X , Y ] withdegreeatmost D + E 1ineachvariablesuchthat P(F , G) = 0.
1.5 Let F bein C[t ] withdegreeatmost D ≥ 1andlet G bein C[t ] with degreeatmost E ≥ 1.Showthatthereis P = 0in C[X , Y ] withdegreeatmost E in X anddegreeatmost DE E + 1in Y suchthat P(F , G) = 0.
1.6 Let t , u beindependentvariables,andlet F , G, H bein C[t , u].Showthat thereis P = 0in C[X , Y , Z ] suchthat P(F , G, H ) = 0.
1.7 Showthatthereisanabsoluteconstant c (thatis,notdependingonany parameters)withthefollowingproperty.Let F bein C[X ] withdegreeatmost D ≥ 2.Thenthereis P = 0in C[X ] withdegreeatmost cD log D suchthat PF hastheform p ap X p ,where p runsoverthesetofprimes.
1.8 Let F bein C[X ].Find P = 0in C[X ] suchthat PF hastheform N n=0 an X 3n .
1.9 Let F bein Fp [X ].Showthatthereis P = 0in Fp [X ] suchthat G = PF satisfies G(X1 + X2 ) = G(X1 ) + G(X2 ) in Fp [X1 , X2 ]
1.10 Let F bein C[t ] withdegreeatmost D ≥ 1andlet G bein C[t ] with degreeatmost E ≥ 1.Showthatthereis P = 0in C[X , Y ] withdegreeatmost E in X anddegreeatmost D in Y suchthat P(F , G) = 0[Hint:resultants].
1.11 Let P = 0in C[X , Y ] besuchthat P(t1948 + t 666 + 1, t 1291 + t 163 + t ) = 0.Showthat P hasdegreeatleast1291in X anddegreeatleast1948in Y (compare Exercise5.7).
1.12 Canoneessentiallyimprovethe D 2 D in Exercise1.2?Idon’tknow.
1.13 Let F , G bein C(t ) (rationalfunctions).Showthatthereis P = 0in C[X , Y ] suchthat P(F , G) = 0.
1.14 Let F = 256 (t 2 t + 1)3 t 2 (t 1)2 , G = 256 (t 2 + t + 1)3 t 2 (t + 1)2 .
Showthat P(F , G) = 0for P = X 3 Y 2X 2 Y 2 + XY 3 1728(X 3 + Y 3 ) + 1216(X 2 Y + XY 2 ) +3538944(X 2 + Y 2 ) 2752512XY 2415919104(X + Y ) + 549755813888.
1.15 Let F = tu(t 10 + 11t 5 u 5 u 10 ), G =−t 20 u 20 + 228(t 15 u 5 t 5 u 15 ) 494t 10 u 10 , H = t 30 + u 30 + 522(t 25 u 5 t 5 u 25 ) 10005(t 20 u 10 + t 10 u 20 )
Showthat G3 + H 2 = 1728F 5 .(Thisisrelatedtotheicosahedron–seeKlein, 1956.)
1.16 Let F = 1728 u3 u3 v2 , G =−1728 u2 v u3 v2 , H =−288 u(tuv 3u3 4v2 ) u3 v2 , K =−24 3t 2 u2 v 18tu4 24tuv2 + 95u3 v + 16v3 u3 v2 .
Itisprovedinanyelementarytextonnumbertheorythat e isirrational; theproofisbasedontherapidconvergenceoftheseriestogetherwiththe reasonablebehaviourofthedenominators.Wegiveaproofnevertheless.
Considerthetruncation
for n = 0,1,2, .Thefirsttermontheextremeright-handsidedominates andindeed
Thus0 < fn < 2/(n + 1)! and
Nowif s isadenominatorfortherational e,thenmultiplyingby s and making n tendtoinfinitygivesacontradictiontotheso-calledFundamental TheoremofTranscendencethateverynon-zerointegerhasabsolutevalueat least1.
Theproofisslightlyeasierforthealternatingseries
becausewenolongerneedthedominance.
Theproofsextendtogivethelinearindependenceover Q of1, e, e 1 ,which amountstothefactthat e cannotbequadraticover Q.Inparticular e2 is irrational.Butthereisaminorsnag.Weassumethat r + se + te 1 = 0for integers r , s, t notallzero,andthenfor
weget
Hencethe n!fn areintegerstendingtozeroas n tendstoinfinity.Butweno longerknowthattheseintegersarenon-zeroasin(2.1).
Infactitisnottoohardtoshowthat
isimpossibleforany n.Namely,
andso
Thus(2.2)wouldimply s = t = 0so r = 0too,acontradiction.
(1 t )n dt (see Exercise2.14).It canalsobedonewithauxiliarypolynomialsofthetypementionedinChapter 1,mostefficientlybyintroducingderivativesasinthedifferentialequation (ez ) = ez .Howeversomeextraarithmeticandanalyticmachineryisneeded. Thiswillbeintroducedstepbystepinthefollowingchapters.Bythesemeans wewillshowinChapter13that
Theproofworksiftheestimatetendstozeroas n tendstoinfinity(assuming wecanruleoutthesnag fn = 0asin(2.2)above).Unfortunatelythisisthe caseonlyfor a = 1.
Another random document with no related content on Scribd:
OR IMPLIED, INCLUDING BUT NOT LIMITED TO WARRANTIES OF MERCHANTABILITY OR FITNESS FOR ANY PURPOSE.
1.F.5. Some states do not allow disclaimers of certain implied warranties or the exclusion or limitation of certain types of damages. If any disclaimer or limitation set forth in this agreement violates the law of the state applicable to this agreement, the agreement shall be interpreted to make the maximum disclaimer or limitation permitted by the applicable state law. The invalidity or unenforceability of any provision of this agreement shall not void the remaining provisions.
1.F.6. INDEMNITY - You agree to indemnify and hold the Foundation, the trademark owner, any agent or employee of the Foundation, anyone providing copies of Project Gutenberg™ electronic works in accordance with this agreement, and any volunteers associated with the production, promotion and distribution of Project Gutenberg™ electronic works, harmless from all liability, costs and expenses, including legal fees, that arise directly or indirectly from any of the following which you do or cause to occur: (a) distribution of this or any Project Gutenberg™ work, (b) alteration, modification, or additions or deletions to any Project Gutenberg™ work, and (c) any Defect you cause.
Section 2. Information about the Mission of Project Gutenberg™
Project Gutenberg™ is synonymous with the free distribution of electronic works in formats readable by the widest variety of computers including obsolete, old, middle-aged and new computers. It exists because of the efforts of hundreds of volunteers and donations from people in all walks of life.
Volunteers and financial support to provide volunteers with the assistance they need are critical to reaching Project
Gutenberg™’s goals and ensuring that the Project Gutenberg™ collection will remain freely available for generations to come. In 2001, the Project Gutenberg Literary Archive Foundation was created to provide a secure and permanent future for Project Gutenberg™ and future generations. To learn more about the Project Gutenberg Literary Archive Foundation and how your efforts and donations can help, see Sections 3 and 4 and the Foundation information page at www.gutenberg.org.
Section 3. Information about the Project Gutenberg Literary Archive Foundation
The Project Gutenberg Literary Archive Foundation is a nonprofit 501(c)(3) educational corporation organized under the laws of the state of Mississippi and granted tax exempt status by the Internal Revenue Service. The Foundation’s EIN or federal tax identification number is 64-6221541. Contributions to the Project Gutenberg Literary Archive Foundation are tax deductible to the full extent permitted by U.S. federal laws and your state’s laws.
The Foundation’s business office is located at 809 North 1500 West, Salt Lake City, UT 84116, (801) 596-1887. Email contact links and up to date contact information can be found at the Foundation’s website and official page at www.gutenberg.org/contact
Section 4. Information about Donations to the Project Gutenberg Literary Archive Foundation
Project Gutenberg™ depends upon and cannot survive without widespread public support and donations to carry out its mission of increasing the number of public domain and licensed works that can be freely distributed in machine-readable form
accessible by the widest array of equipment including outdated equipment. Many small donations ($1 to $5,000) are particularly important to maintaining tax exempt status with the IRS.
The Foundation is committed to complying with the laws regulating charities and charitable donations in all 50 states of the United States. Compliance requirements are not uniform and it takes a considerable effort, much paperwork and many fees to meet and keep up with these requirements. We do not solicit donations in locations where we have not received written confirmation of compliance. To SEND DONATIONS or determine the status of compliance for any particular state visit www.gutenberg.org/donate.
While we cannot and do not solicit contributions from states where we have not met the solicitation requirements, we know of no prohibition against accepting unsolicited donations from donors in such states who approach us with offers to donate.
International donations are gratefully accepted, but we cannot make any statements concerning tax treatment of donations received from outside the United States. U.S. laws alone swamp our small staff.
Please check the Project Gutenberg web pages for current donation methods and addresses. Donations are accepted in a number of other ways including checks, online payments and credit card donations. To donate, please visit: www.gutenberg.org/donate.
Section 5. General Information About Project Gutenberg™ electronic works
Professor Michael S. Hart was the originator of the Project Gutenberg™ concept of a library of electronic works that could be freely shared with anyone. For forty years, he produced and
distributed Project Gutenberg™ eBooks with only a loose network of volunteer support.
Project Gutenberg™ eBooks are often created from several printed editions, all of which are confirmed as not protected by copyright in the U.S. unless a copyright notice is included. Thus, we do not necessarily keep eBooks in compliance with any particular paper edition.
Most people start at our website which has the main PG search facility: www.gutenberg.org.
This website includes information about Project Gutenberg™, including how to make donations to the Project Gutenberg Literary Archive Foundation, how to help produce our new eBooks, and how to subscribe to our email newsletter to hear about new eBooks.
