數(shù)論IV:超越數(shù)

前言

要使我國(guó)的數(shù)學(xué)事業(yè)更好地發(fā)展起來(lái)，需要數(shù)學(xué)家淡泊名利并付出更艱苦地努力。另一方面，我們也要從客觀上為數(shù)學(xué)家創(chuàng)造更有利的發(fā)展數(shù)學(xué)事業(yè)的外部環(huán)境，這主要是加強(qiáng)對(duì)數(shù)學(xué)事業(yè)的支持與投資力度，使數(shù)學(xué)家有較好的工作與生活條件，其中也包括改善與加強(qiáng)數(shù)學(xué)的出版工作。從出版方面來(lái)講，除了較好較快地出版我們自己的成果外，引進(jìn)國(guó)外的先進(jìn)出版物無(wú)疑也是十分重要與必不可少的。從數(shù)學(xué)來(lái)說(shuō)，施普林格（springer）出版社至今仍然是世界上最具權(quán)威的出版社?？茖W(xué)出版社影印一批他們出版的好的新書，使我國(guó)廣大數(shù)學(xué)家能以較低的價(jià)格購(gòu)買，特別是在邊遠(yuǎn)地區(qū)工作的數(shù)學(xué)家能普遍見到這些書，無(wú)疑是對(duì)推動(dòng)我國(guó)數(shù)學(xué)的科研與教學(xué)十分有益的事。這次科學(xué)出版社購(gòu)買了版權(quán)，一次影印了23本施普林格出版社出版的數(shù)學(xué)書，就是一件好事，也是值得繼續(xù)做下去的事情。大體上分一下，這23本書中，包括基礎(chǔ)數(shù)學(xué)書5本，應(yīng)用數(shù)學(xué)書6本與計(jì)算數(shù)學(xué)書12本，其中有些書也具有交叉性質(zhì)。這些書都是很新的，2000年以后出版的占絕大部分，共計(jì)16本，其余的也是1990年以后出版的。這些書可以使讀者較快地了解數(shù)學(xué)某方面的前沿，例如基礎(chǔ)數(shù)學(xué)中的數(shù)論、代數(shù)與拓?fù)淙?，都是由該領(lǐng)域大數(shù)學(xué)家編著的“數(shù)學(xué)百科全書”的分冊(cè)。對(duì)從事這方面研究的數(shù)學(xué)家了解該領(lǐng)域的前沿與全貌很有幫助。按照學(xué)科的特點(diǎn)，基礎(chǔ)數(shù)學(xué)類的書以“經(jīng)典”為主，應(yīng)用和計(jì)算數(shù)學(xué)類的書以“前沿”為主。這些書的作者多數(shù)是國(guó)際知名的大數(shù)學(xué)家，例如《拓?fù)鋵W(xué)》一書的作者諾維科夫是俄羅斯科學(xué)院的院士，曾獲“菲爾茲獎(jiǎng)”和“沃爾夫數(shù)學(xué)獎(jiǎng)”。這些大數(shù)學(xué)家的著作無(wú)疑將會(huì)對(duì)我國(guó)的科研人員起到非常好的指導(dǎo)作用。當(dāng)然，23本書只能涵蓋數(shù)學(xué)的一部分，所以，這項(xiàng)工作還應(yīng)該繼續(xù)做下去。更進(jìn)一步，有些讀者面較廣的好書還應(yīng)該翻譯成中文出版，使之有更大的讀者群?？傊?，我對(duì)科學(xué)出版社影印施普林格出版社的部分?jǐn)?shù)學(xué)著作這一舉措表示熱烈的支持，并盼望這一工作取得更大的成績(jī)。

內(nèi)容概要

This book is a survey of the most important directions of research in transcendental number theory. The central topics in this theory include proofs of irrationality and transcendence of various numbers，especially those，that arise as the values of special functions. Questions of this sort go back to ancient times. An example is the old problem of squaring the circle，which Lindemann showed to bc impossible in 1882，when hc proved that Pi is a trandental number. Euler's conjecture that the logarithm of an algebraic number to an algebraic base is transcendental was included in Hilbert's famous list of open problems; this conjecture was proved by Gel'fond and Schneider in 1934. A more recent result was Anerv's surprising proof of the irrationality of ξ(3)in 1979.    The quantitative aspects of the theory have important applications to the study of Diophantine equations and other areas of number theory. For a reader interested in different branches of number theory，this monograph provides both an overview of the central ideas and techniques of transcendental number theory，and also a guide to the most important results and references.

作者簡(jiǎn)介

作者：(俄羅斯)帕爾申 (Parshin.A.N.) (俄羅斯)I.R.Shafarevich

書籍目錄

NotationIntroduction  0.1 Preliminary Remarks  0.2 Irrationality of 2  0.3 The Number π  0.4 Transcendental Numbers  0.5 Approximation of Algebraic Numbers  0.6 Transcendence Questions and Other Branches of Number Theory  0.7 The Basic Problems Studied in Transcendental Number Theory  0.8 Different Ways of Giving the Numbers  0.9 MethodsChapter 1 Approximation of Algebraic Numbers　1 Preliminaries  　1.1 Parameters for Algebraic Numbers and Polynomials    1.2 Statement of the Problem    1.3 Approximation of Rational Numbers    1.4 Continued Fractions    1.5 Quadratic Irrationalities    1.6 Liouville's Theorem and Liouville Numbers    1.7 Generalization of Liouville's Theorem　2 Approximations of Algebraic Numbers and Thue's Equation    2.1 Thue's Equation    2.2 The Case n = 2    2.3 The Case n > 3　3 Strengthening Liouville's Theorem First Version of Thue's Method    3.1 A Way to Bound qθ-ρ    3.2 Construction of Rational Approximations for    3.3 Thue's First Result    3.4 Effectiveness    3.5 Effective Analogues of Theorem 1.6    3.6 The First Effective Inequalities of Baker    3.7 Effective Bounds on Linear Forms in Algebraic Numbers　4 Stronger and More General Versions of Liouville's Theorem and Thue's Theorem    4.1 The Dirichlet Pigeonhole Principle    4.2 Thue's Method in the General Case    4.3 Thue's Theorem on Approximation of Algebraic Numbers    4.4 The Non-effectiveness of Thue's Theorems　5 Further Development of Thue's Method    5.1 Siegel's Theorem    5.2 The Theorems of Dyson and Gel'fond    5.3 Dyson's Lemma    5.4 Bombieri's Theorem　6 Multidimensional Variants of the Thue-Siegel Method    6.1 Preliminary Remarks    6.2 Siegel's Theorem    6.3 The Theorems of Schneider and Mahler  7 Roth's Theorem    7.1 Statement of the Theorem    7.2 The Index of a Polynomial    7.3 Outline of the Proof of Roth's Theorem    7.4 Approximation of Algebraic Numbers by Algebraic Numbers    7.5 The Number k in Roth's Theorem    7.6 Approximation by Numbers of a Special Type    7.7 Transcendence of Certain Numbers    7.8 The Number of Solutions to the Inequality (62) and Certain Diophantine Equations  8 Linear Forms in Algebraic Numbers and Schmidt's Theorem    8.1 Elementary Estimates    8.2 Schmidt's Theorem    8.3 Minkowski's Theorem on Linear Forms    8.4 Schmidt's Subspace TheoremChapter 2 Effective Constructions in Transcendental Number TheoryChapter 3 Hillbert's Seventh ProblemChapter 4 Multidimensional Generalization of Hillbert's Seventh ProblemChapter 5 Values of Analytic Functions That Satisgy Linear Differential EquationsChapter 6 Algebraic Independence of the Values of Analytic Functions That Have an Additaon La

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