A Treatise on Algebraic Plane Curves

Julian Lowell Coolidge

Courier Corporation, Jan 1, 2004 - Mathematics - 513 pages

Students and teachers will welcome the return of this unabridged reprint of one of the first English-language texts to offer full coverage of algebraic plane curves. It offers advanced students a detailed, thorough introduction and background to the theory of algebraic plane curves and their relations to various fields of geometry and analysis.
The text treats such topics as the topological properties of curves, the Riemann-Roch theorem, and all aspects of a wide variety of curves including real, covariant, polar, containing series of a given sort, elliptic, hyperelliptic, polygonal, reducible, rational, the pencil, two-parameter nets, the Laguerre net, and nonlinear systems of curves. It is almost entirely confined to the properties of the general curve rather than a detailed study of curves of the third or fourth order. The text chiefly employs algebraic procedure, with large portions written according to the spirit and methods of the Italian geometers. Geometric methods are much employed, however, especially those involving the projective geometry of hyperspace.
Readers will find this volume ample preparation for the symbolic notation of Aronhold and Clebsch.

Contents

• ELEMENTARY THEORY: 1
• Resultant of two polynomials: 7
• CHAPTER V: 12
• CHAPTER II: 14
• Tangential equations: 22
• Number of points needed to determine a curve of given order: 29
• CHAPTER III: 39
• Parts of curves: 50
• 559: 57
• Cross ratios: 69
• Polar operator: 75
• Conditions for a singular point of given order: 86
• Lefschetzs theorem about nodes: 104
• Genus: 108
• Definition of genus its importance: 121
• Deduction of the general formula Riemanns theorem: 128
• Correspondences on different curves: 134
• First polars: 140
• Simplest linear systems: 147
• Singular points of Hessian: 153
• 71: 154
• Satellite curve: 160
• Warings theorem: 166
• Products of distances: 174
• Sums of angles determined by tangents and foci: 180
• Contacts with asymptotes: 181
• 74: 185
• 14: 188
• 20: 194
• Limiting cases: 200
• Nöthers transformation theorem: 207
• General series development of a branch: 213
• Order and class of a branch: 219
• Number of intersections in general case: 225
• CHAPTER III: 232
• Satellite points: 239
• Residuation theorems: 245
• Adjunction theorem: 254
• Representation in hyperspace: 257
• 23: 260
• Canonical series and canonical curves: 267
• Sufficiency of these conditions: 273
• Integrals of other sorts: 275
• Groups common to a g and a g: 281
• CHAPTER IV: 289
• Number of parameters determining a special gN: 298
• Two types of birational transformation of a general cubic: 304
• Transformation of hyperelliptic curve to canonical form: 305
• Reducible curves: 312
• Determination of the number when: 320
• CHAPTER VII: 329
• Relation of circuits to rational points in hyperspace: 336
• p p correspondences: 342
• Curves with only simple branches Clebschs transformation: 348
• CHAPTER VIII: 354
• Applications of uniformization: 360
• Symbolic notation: 368
• Determination of the equation of a rational curve: 370
• Cuspidal and undulational conditions: 376
• Postulation by means of singular points: 383
• Situation of singular points: 389
• Transformation of the pure adjoint system: 395
• Reduction of singularities: 399
• Reduction of curves lacking adjoint systems of high index: 406
• Formation of an invariant operator: 412
• The inflexional locus: 420
• General expression for a nonlinear system: 425
• The inflexions: 433
• Number of curves in a kparameter system which touch curves: 439
• Nöthers factorization theorem: 446
• Montesanos theorem: 453
• Transformations in one plane: 459
• Fixed points of two sorts: 466
• Transformations with curves of fixed points: 474
• Involutory transformations of lowest class: 482
• CHAPTER VIII: 489
• Finite groups: 496
• Comparison of general form with power of a linear form: 501
• 81: 511
• 329: 512

Bibliographic information

• Title: A Treatise on Algebraic Plane Curves; Volume 1 of Dover Books on Mathematics Series; Dover phoenix editions; Volume 1 of Phoenix Edition Series
• Author: Julian Lowell Coolidge
• Edition: illustrated, reprint
• Publisher: Courier Corporation, 2004
• ISBN: 0486495760, 9780486495767
• Length: 513 pages
• Subjects: Mathematics / Algebra / General; Mathematics / Geometry / Algebraic