A guide to plane algebraic curves:
Saved in:
| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Washington, DC
MAA
2011
|
| Series: | The Dolciani mathematical expositions
46 MAA guides 7 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XV, 193 S. graph. Darst. |
| ISBN: | 9780883853535 |
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MARC
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| 245 | 1 | 0 | |a A guide to plane algebraic curves |c Keith Kendig |
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Record in the Search Index
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| adam_text | Titel: A guide to plane algebraic curves
Autor: Kendig, Keith
Jahr: 2011
Contents
Preface......................................................... vii
1 A Gallery of Algebraic Curves............................... 1
1.1 Curves of Degree One and Two............... 1
1.2 Curves of Degree Three and Higher............ 4
1.3 Six Basic Cubics...................... 7
1.4 Some Curves in Polar Coordinates............. 7
1.5 Parametric Curves...................... 12
1.6 TheResultant........................ 13
1.7 Back to an Example..................... 15
1.8 Lissajous Figures...................... 16
1.9 Morphing Between Curves................. 18
1.10 Designer Curves....................... 22
2 Points at Infinity............................................. 29
2.1 Adjoining Points at Infinity................. 30
2.2 Examples.......................... 33
2.3 A Basic Picture....................... 35
2.4 Basic Definitions...................... 37
2.5 Further Examples...................... 40
3 From Real to Complex....................................... 45
3.1 Definitions.......................... 46
3.2 The IdeaofMultiplicity; Examples............. 47
3.3 A Reality Check....................... 52
3.4 A Factorization Theorem for Polynomials in C[x,v] ... 54
3.5 Local Parametrizations of a Plane Algebraic Curve .... 55
3.6 Definition of Intersection
Multiplicity for Two Branches............... 56
3.7 An Example......................... 57
XIII
XIV CONTENTS
3.8 Multiplicity at an Intersection Point of Two Plane Algebraic
Curves............................ 58
3.9 Intersection Multiplicity Without Parametrizations..... 59
3.10 Bezout s theorem...................... 63
3.11 Bezout s theorem Generalizes the Fundamental Theorem of
Algebra........................... 69
3.12 An Application of Bezout s theorem: Pascal s theorem . . 71
4 Topologyof Algebraic Curves in P2(C)....................... 75
4.1 Introduction......................... 75
4.2 Connectedness ....................... 76
4.3 Algebraic Curves are Connected.............. 77
4.4 Orientable Two-Manifolds................. 84
4.5 Nonsingular Curves are Two-Manifolds.......... 86
4.6 Algebraic Curves are Orientable.............. 87
4.7 The Genus Formula..................... 88
5 Singularities................................................ 93
5.1 Introduction......................... 93
5.2 Definitions and Examples.................. 94
5.3 Singularities at Infinity................... 97
5.4 Nonsingular Projective Curves............... 97
5.5 Singularities and Polynomial Degree............ 99
5.6 Singularities and Genus................... 102
5.7 A More General Genus Formula.............. 110
5.8 Non-Ordinary Singularities................. 111
5.9 Further Examples...................... 115
5.10 Singularities versus Doing Math on Curves ........ 117
5.11 The Function Field of an Irreducible Curve........ 118
5.12 Birational Equivalence................... 119
5.13 Examples of Birational Equivalence............ 121
5.14 Space-Curve Models.................... 127
5.15 Resolving a Higher-Order Ordinary Singularity...... 130
5.16 Examples of Resolving an Ordinary Singularity...... 131
5.17 Resolving Several Ordinary Singularities.......... 137
5.18 Quadratic Transformations................. 138
6 The Big Three: C,K,S.......................................143
6.1 Function Fields....................... 145
6.2 Compact Riemann Surfaces................. 146
6.3 Projective Plane Curves................... 152
CONTENTS XV
6-4 /i, fz, f: Curves and Function Fields........... 153
6.5 gi, g2, g: Compact Riemann Surfaces and Curves..... 155
6.6 hi, /i2 h: Function Fields and Compact Riemann Surfaces 156
6.7 Genus............................ 156
6.8 Genus 0........................... 157
6.9 Genus One ......................... 158
6.10 AnAnalogy......................... 166
6.11 Equipotentials and Streamlines............... 170
6.12 Differentials Generate Vector Fields............ 174
6.13 A Major Difference..................... 175
6.14 Divisors........................... 179
6.15 The Riemann-Roch theorem................ 182
Bibliography...................................................185
Index...........................................................189
About the Author...............................................193
|
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| id | DE-604.BV040133301 |
| illustrated | Illustrated |
| indexdate | 2024-07-10T00:17:38Z |
| institution | BVB |
| isbn | 9780883853535 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-024990403 |
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| physical | XV, 193 S. graph. Darst. |
| publishDate | 2011 |
| publishDateSearch | 2011 |
| publishDateSort | 2011 |
| publisher | MAA |
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| series | The Dolciani mathematical expositions MAA guides |
| series2 | The Dolciani mathematical expositions MAA guides |
| spelling | Kendig, Keith 1938- Verfasser (DE-588)1029541809 aut A guide to plane algebraic curves Keith Kendig Plane algebraic curves Washington, DC MAA 2011 XV, 193 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier The Dolciani mathematical expositions 46 MAA guides 7 Ebene algebraische Kurve (DE-588)4124805-3 gnd rswk-swf Algebraische Geometrie (DE-588)4001161-6 gnd rswk-swf Algebraische Geometrie (DE-588)4001161-6 s Ebene algebraische Kurve (DE-588)4124805-3 s DE-604 The Dolciani mathematical expositions 46 (DE-604)BV001900740 46 MAA guides 7 (DE-604)BV035218151 7 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=024990403&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Kendig, Keith 1938- A guide to plane algebraic curves The Dolciani mathematical expositions MAA guides Ebene algebraische Kurve (DE-588)4124805-3 gnd Algebraische Geometrie (DE-588)4001161-6 gnd |
| subject_GND | (DE-588)4124805-3 (DE-588)4001161-6 |
| title | A guide to plane algebraic curves |
| title_alt | Plane algebraic curves |
| title_auth | A guide to plane algebraic curves |
| title_exact_search | A guide to plane algebraic curves |
| title_full | A guide to plane algebraic curves Keith Kendig |
| title_fullStr | A guide to plane algebraic curves Keith Kendig |
| title_full_unstemmed | A guide to plane algebraic curves Keith Kendig |
| title_short | A guide to plane algebraic curves |
| title_sort | a guide to plane algebraic curves |
| topic | Ebene algebraische Kurve (DE-588)4124805-3 gnd Algebraische Geometrie (DE-588)4001161-6 gnd |
| topic_facet | Ebene algebraische Kurve Algebraische Geometrie |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=024990403&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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