数论IV:超越数

图书标准信息

• 作者 [俄罗斯]帕尔甲 著
• 出版社 科学出版社
• 出版时间 2009-01
• 版次 1
• ISBN 9787030235084
• 定价 80.00元
• 装帧 精装
• 开本 16开
• 纸张 胶版纸
• 页数 345页
• 字数 435千字
• 正文语种 英语

【内容简介】

　　《数论4：超越数(影印版)》isasurveyofthemostimportantdirectionsofresearchintranscendentalnumbertheory.Thecentraltopicsinthistheoryincludeproofsofirrationalityandtranscendenceofvariousnumbers，especiallythose，thatariseasthevaluesofspecialfunctions.Questionsofthissortgobacktoancienttimes.Anexampleistheoldproblemofsquaringthecircle，whichLindemannshowedtobcimpossiblein1882，whenhcprovedthatPiisatrandentalnumber.EulersconjecturethatthelogarithmofanalgebraicnumbertoanalgebraicbaseistranscendentalwasincludedinHilbertsfamouslistofopenproblems;thisconjecturewasprovedbyGelfondandSchneiderin1934.AmorerecentresultwasAnervssurprisingproofoftheirrationalityofξ(3)in1979.
　　ThequantitativeaspectsofthetheoryhaveimportantapplicationstothestudyofDiophantineequationsandotherareasofnumbertheory.Forareaderinterestedindifferentbranchesofnumbertheory，thismonographprovidesbothanoverviewofthecentralideasandtechniquesoftranscendentalnumbertheory，andalsoaguidetothemostimportantresultsandreferences.

【目录】

Notation
Introduction
0.1PreliminaryRemarks
0.2Irrationalityof2
0.3TheNumberπ
0.4TranscendentalNumbers
0.5ApproximationofAlgebraicNumbers
0.6TranscendenceQuestionsandOtherBranchesofNumberTheory
0.7TheBasicProblemsStudiedinTranscendentalNumberTheory
0.8DifferentWaysofGivingtheNumbers
0.9Methods
Chapter1ApproximationofAlgebraicNumbers
1Preliminaries
　1.1ParametersforAlgebraicNumbersandPolynomials
1.2StatementoftheProblem
1.3ApproximationofRationalNumbers
1.4ContinuedFractions
1.5QuadraticIrrationalities
1.6LiouvillesTheoremandLiouvilleNumbers
1.7GeneralizationofLiouvillesTheorem
2ApproximationsofAlgebraicNumbersandThuesEquation
2.1ThuesEquation
2.2TheCasen=2
2.3TheCasen>3
3StrengtheningLiouvillesTheoremFirstVersionofThuesMethod
3.1AWaytoBoundqθ-ρ
3.2ConstructionofRationalApproximationsfor
3.3ThuesFirstResult
3.4Effectiveness
3.5EffectiveAnaloguesofTheorem1.6
3.6TheFirstEffectiveInequalitiesofBaker
3.7EffectiveBoundsonLinearFormsinAlgebraicNumbers
4StrongerandMoreGeneralVersionsofLiouvillesTheoremandThuesTheorem
4.1TheDirichletPigeonholePrinciple
4.2ThuesMethodintheGeneralCase
4.3ThuesTheoremonApproximationofAlgebraicNumbers
4.4TheNon-effectivenessofThuesTheorems
5FurtherDevelopmentofThuesMethod
5.1SiegelsTheorem
5.2TheTheoremsofDysonandGelfond
5.3DysonsLemma
5.4BombierisTheorem
6MultidimensionalVariantsoftheThue-SiegelMethod
6.1PreliminaryRemarks
6.2SiegelsTheorem
6.3TheTheoremsofSchneiderandMahler
7RothsTheorem
7.1StatementoftheTheorem
7.2TheIndexofaPolynomial
7.3OutlineoftheProofofRothsTheorem
7.4ApproximationofAlgebraicNumbersbyAlgebraicNumbers
7.5TheNumberkinRothsTheorem
7.6ApproximationbyNumbersofaSpecialType
7.7TranscendenceofCertainNumbers
7.8TheNumberofSolutionstotheInequality(62)andCertainDiophantineEquations
8LinearFormsinAlgebraicNumbersandSchmidtsTheorem
8.1ElementaryEstimates
8.2SchmidtsTheorem
8.3MinkowskisTheoremonLinearForms
8.4SchmidtsSubspaceTheorem
Chapter2EffectiveConstructionsinTranscendentalNumberTheory
Chapter3HillbertsSeventhProblem
Chapter4MultidimensionalGeneralizationofHillbertsSeventhProblem
Chapter5ValuesofAnalyticFunctionsThatSatisgyLinearDifferentialEquations
Chapter6AlgebraicIndependenceoftheValuesofAnalyticFunctionsThatHaveanAdditaonLa