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代數曲線幾何初步 版權信息
- ISBN：9787506292641
- 條形碼：9787506292641 ; 978-7-5062-9264-1
- 裝幀：一般膠版紙
- 冊數：暫無
- 重量：暫無
- 所屬分類：>>
代數曲線幾何初步 內容簡介
general background i first became involved in the teaching of geometry about twenty years ago，when my department introduced an optional second year course on the geometry of plane curves，partly to redress the imbalance in the teaching of the subject。it was mildly revolutionary，since it went back to an earlier sct of precepts where the differential and algebraic geometry of cuwes were pursued simultaneously,to their mutua！advantage.
代數曲線幾何初步 目錄
list of illustrations
list of tables
preface
1 real algebraic curves
1.1 parametrized and implicit curves
1.2 introductory examples
1.3 curves in planar kinematics
2 general ground fields
2.1 two motivating examples
2.2 groups, rings and fields
2.3 general affine planes and curves
2.4 zero sets of algebraic curves
3 polynomial algebra
3.1 factorization in domains
3.2 polynomials in one variable
3.3 polynomials in several variables
3.4 homogeneous polynomials
3.5 formal differentiation
4 atfine equivalence
4.1 affine maps
4.2 affline equivalent curves
4.3 degree as an affine invariant
4.4 centres as affine invariants
5 affline conics
5.1 affline classification
5.2 the delta invariants
5.3 uniqueness of equations
6 singularities of afline curves
6.1 intersection numbers
6.2 multiplicity of a point on a curve
6.3 singular points
7 tangents to afline curves
7.1 generalities about tangents
7.2 tangents at simple points
7.3 tangents at double points
7.4 tangents at points of higher multiplicity
8 rational afline curves
8.1 rational curves
8.2 diophantine equations
8.3 conics and integrals
9 projective algebraic curves
9.1 the projective plane
9.2 projective lines
9.3 atfine planes in the projective plane
9.4 projective curves
9.5 affine views of projective curves
10 singularities of projective curves
10.1 intersection numbers
10.2 multiplicity of a point on a curve
10.3 singular points
10.4 delta invariants viewed projectively
11 projective equivalence
11.1 projective maps
11.2 projective equivalence
11.3 projective conics
11.4 afline and projective equivalence
12 projective tangents
12.1 tangents to projective curves
12.2 tangents at simple points
12.3 centres viewed projectively
12.4 foci viewed projectively
12.5 tangents at singular points
12.6 asymptotes
13 flexes
13.1 hessian curves
13.2 configurations of flexes
14 intersections of proiective curves
14．1 the geometric idea
14．2 resultants in one variable
14．3 resultants in severa!variables
14．4 b6zout’s theorem
14．5 thc multiplicity inequality
14．6 invariance of the intersection number
15 proiective cubics
15．1 geometric types 0f cubics
15．2 cubics of general type
15．3 singular irreducible cubics
15．4 reducible cubics
16 linear systems
16．1 projective spaces of curves
16．2 pcncils of cuines
16．3 solving quartic equations
16．4 subspaces or projective spaces
16．5 linear systems of culwes
16．6 dual culnes
17 the group structure on a cubic
17．1 the nine associated points
17．2 the star operation
17．3 cubics as groups
17．4 group computations
17．5 determination of the groups
18 rational projective curves
18．1 thc projective concept
18．2 quartics with three double points
18．3 thc deficiency of a chive
18．4 some rational curves
18．5 some non-rational curves
index
list of tables
preface
1 real algebraic curves
1.1 parametrized and implicit curves
1.2 introductory examples
1.3 curves in planar kinematics
2 general ground fields
2.1 two motivating examples
2.2 groups, rings and fields
2.3 general affine planes and curves
2.4 zero sets of algebraic curves
3 polynomial algebra
3.1 factorization in domains
3.2 polynomials in one variable
3.3 polynomials in several variables
3.4 homogeneous polynomials
3.5 formal differentiation
4 atfine equivalence
4.1 affine maps
4.2 affline equivalent curves
4.3 degree as an affine invariant
4.4 centres as affine invariants
5 affline conics
5.1 affline classification
5.2 the delta invariants
5.3 uniqueness of equations
6 singularities of afline curves
6.1 intersection numbers
6.2 multiplicity of a point on a curve
6.3 singular points
7 tangents to afline curves
7.1 generalities about tangents
7.2 tangents at simple points
7.3 tangents at double points
7.4 tangents at points of higher multiplicity
8 rational afline curves
8.1 rational curves
8.2 diophantine equations
8.3 conics and integrals
9 projective algebraic curves
9.1 the projective plane
9.2 projective lines
9.3 atfine planes in the projective plane
9.4 projective curves
9.5 affine views of projective curves
10 singularities of projective curves
10.1 intersection numbers
10.2 multiplicity of a point on a curve
10.3 singular points
10.4 delta invariants viewed projectively
11 projective equivalence
11.1 projective maps
11.2 projective equivalence
11.3 projective conics
11.4 afline and projective equivalence
12 projective tangents
12.1 tangents to projective curves
12.2 tangents at simple points
12.3 centres viewed projectively
12.4 foci viewed projectively
12.5 tangents at singular points
12.6 asymptotes
13 flexes
13.1 hessian curves
13.2 configurations of flexes
14 intersections of proiective curves
14．1 the geometric idea
14．2 resultants in one variable
14．3 resultants in severa!variables
14．4 b6zout’s theorem
14．5 thc multiplicity inequality
14．6 invariance of the intersection number
15 proiective cubics
15．1 geometric types 0f cubics
15．2 cubics of general type
15．3 singular irreducible cubics
15．4 reducible cubics
16 linear systems
16．1 projective spaces of curves
16．2 pcncils of cuines
16．3 solving quartic equations
16．4 subspaces or projective spaces
16．5 linear systems of culwes
16．6 dual culnes
17 the group structure on a cubic
17．1 the nine associated points
17．2 the star operation
17．3 cubics as groups
17．4 group computations
17．5 determination of the groups
18 rational projective curves
18．1 thc projective concept
18．2 quartics with three double points
18．3 thc deficiency of a chive
18．4 some rational curves
18．5 some non-rational curves
index
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