計算不變量理論

前言

要使我國的數(shù)學(xué)事業(yè)更好地發(fā)展起來，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。從出版方面來講，除了較好較快地出版我們自己的成果外，引進(jìn)國外的先進(jìn)出版物無疑也是十分重要與必不可少的。從數(shù)學(xué)來說，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，使我國廣大數(shù)學(xué)家能以較低的價格購買，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，無疑是對推動我國數(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荆际怯稍擃I(lǐng)域大數(shù)學(xué)家編著的“數(shù)學(xué)百科全書”的分冊。對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，基礎(chǔ)數(shù)學(xué)類的書以“經(jīng)典”為主，應(yīng)用和計算數(shù)學(xué)類的書以“前沿”為主。這些書的作者多數(shù)是國際知名的大數(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”和“沃爾夫數(shù)學(xué)獎”。這些大數(shù)學(xué)家的著作無疑將會對我國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。總之，我對科學(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。

內(nèi)容概要

This book is about the computational aspects of invariant theory.Of central interest is the question how the invariant ring of a given group action can be calculated. Algorithms for this purpose form the main pillars around which the book is built. There are two introductory chapters, one on GrObner basis methods and one on the basic concepts of invariant theory, which prepare the ground for the algorithms. Then algorithms for computing invariants of finite and reductive groups are discussed. Particular emphasis lies on interrelations between structural properties of invariant rings and computational methods. Finally, the book contains a chapter on applications of invariant theory, covering fields as disparate as graph theory, coding theory, dynamical systems, and computer vision.    The book is intended for postgraduate students as well as researchers in geometry, computer algebra, and, of course, invariant theory. The text is enriched with numerous explicit examples which illustrate the theoiw and should be of more than passing interest.

作者簡介

作者：(美國)德克森 (harm derksen)

書籍目錄

Introduction1  Constructive Ideal Theory  1.1  Ideals and GrSbner Bases  1.2  Elimination Ideals    1.3  Syzygy Modules  1.4  Hilbert Series  1.5  The Radical Ideal  1.6  Normalization2  Invariant Theory  2.1  Invariant Rings  2.2  Reductive Groups  2.3  Categorical Quotients  2.4  Homogeneous Systems of Parameters  2.5  The Cohen-Macaulay Property of Invariant Rings  2.6  Hilbert Series of Invariant Rings3  Invariant Theory of Finite Groups  3.1  Homogeneous Components  3.2  Molien's Formula..  3.3  Primary Invariants  3.4  Cohen-Macaulayness  3.5  Secondary Invariants  3.6  Minimal Algebra Generators and Syzygies  3.7  Properties of Invariant Rings  3.8  Noether's Degree Bound  3.9  Degree Bounds in the Modular Case  3.10 Permutation Groups  3.11 Ad Hoc Methods4  Invariant Theory of Reductive Groups  4.1  Computing Invariants of Linearly Reductive Groups   4.2  Improvements and Generalizations  4.3  Invariants of Tori  4.4  Invariants of SLn and GLn  4.5  The Reynolds Operator  4.6  Computing Hilbert Series  4.7  Degree Bounds for Invariants  4.8  Properties of Invariant Rings5  Applications of Invariant Theory  5.1  Cohomology of Finite Groups  5.2  Galois Group Computation  5.3  Noether's Problem and Generic Polynomials  5.4  Systems of Algebraic Equations with Symmetries  5.5  Graph Theory  5.6  Combinatorics  5.7  Coding Theory  5.8  Equivariant Dynamical Systems  5.9  Material Science  5.10  Computer VisionA  Linear Algebraic Groups  A.1  Linear Algebraic Groups  A.2  The Lie Algebra of a Linear Algebraic Group   A.3  Reductive and Semi-simple Groups  A.4  Roots  A.5  Representation TheoryReferencesNotationIndex

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