平面代数曲线

内容概要

Egbert Brieskorn和Horst Knorrer是国际知名学者，在数学界享有盛誉。本书凝聚了作者多年科研和教学成果，适用于科研工作者、高校教师和研究生。

书籍目录

Ⅰ History of algebraic curyres

1.Origin and generation of curves

1.1 The circle and the straight line

1.2 The classical problems of antiquity

1.3 The conic sections

1.4 The cissoid of Diocles

1.5 The conchoid of Nicomedes

1.6 The spiric sections of Perseus

1.7 From the epicycles of Hipparchos to the Wankel motor

1.8 Caustics and contour curves in optics and perspective

1.9 Further examples of curves from science and technology

2.Synthetic and analytic geometry

2.1 Coordinates

2.2 The development of analytic geometry

2.3 Equations for curves

2.4 Examples of the application of analytic methods

2.5 Newton's investigation of cubic curves

3.The development of projective geometry

3.1 Descriptive geometry and projective geometry

3.2 The development of analytic projective geometry

3.3 The projective plane as a manifold

3.4 Complex projective cjeometry

Ⅱ Investigation of curves by elementary algebraic methods

4.Polynomials

4.1 Decomposition into prime factors

4.2 Divisibility properties of polynomials

4.3 Zeroes of polynomials

4.4 Homogeneous and inhomocjeneous polynomials

5.Definition and elementary properties of plane alqebraic curves

5.1 Decomposition into irreducible components

5.2 Intersection of a curve by a line

5.3 Singular points of plane curves

6.The intersection of plane curves

6.1 Bezout's theorem

6.2 Applications of Bezout's theorem

6.3 The intersection ring of P2

7.Some simple types of curves

7.1 Quadrics

7.2 Linear systems of cubics

7.3 Inflection point figures and normal forms of cubics

7.4 Cubics， elliptic curves and abelian varieties

Ⅲ Investigation of curves by resolution of singularities

8.Local investicjations

8.1 Localisation—local rings

8.2 Singularities as analytic set germs

8.3 Newton polycjons and Puiseux expansions

8.4 Resolution of sincjularities by quadratic transformations

8.5 Topology of sinqularities

9.Global investigations

9.1 The Plucker formulae

9.2 The formulaa of Clebsch and Noether

9.3 Differential forms on Riemann surfaces and their periods

Bibliography

Index

作者简介

作者在详细全面地介绍了平面代数理论，并从两方面分析了这个数学的经典研究领域：其在古希腊数学研究中的显著地位；它依然是当代数学研究领域里的灵感激发者和主题。同时该书也为我们综合理解和研究当代关于奇异性的研究打下了基础。第一章中展示了许多拥有优美几何体的特殊曲线——丰富的插图是该书的一大特点，还介绍了投影几何学（在复数域上）。第二章中对Bezout定理进行了简单的证明并详细论述了三次曲线。

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