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Cox D., Little J., O'Shea D. — Ideals, varieties, and algorithms
Cox D., Little J., O'Shea D. — Ideals, varieties, and algorithms



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Название: Ideals, varieties, and algorithms

Авторы: Cox D., Little J., O'Shea D.

Аннотация:

Algebraic Geometry is the study of systems of polynomial equations in one or more variables, asking such questions as: Does the system have finitely many solutions, and if so how can one find them? And if there are infinitely many solutions, how can they be described and manipulated? The solutions of a system of polynomial equations form a geometric object called a variety; the corresponding algebraic object is an ideal. There is a close relationship between ideals and varieties which reveals the intimate link between algebra and geometry. Written at a level appropriate to undergraduates, this book covers such topics as the Hilbert Basis Theorem, the Nullstellensatz, invariant theory, projective geometry, and dimension theory.

The algorithms to answer questions such as those posed above are an important part of algebraic geometry. This book bases its discussion of algorithms on a generalization of the division algorithm for polynomials in one variable that was only discovered it the 1960's. Although the algorithmic roots of algebraic geometry are old, the computational aspects were neglected earlier in this century. This has changed in recent years, and new algorithms, coupled with the power of fast computers, have let to some interesting applications, for example in robotics and in geometric Theorem proving.

In preparing a new edition of "Ideals, Varieties and Algorithms" the authors present an improved proof of the Buchberger Criterion as well as a proof of Bezout's Theorem. Appendix C contains a new section on Axiom and an update about Maple, Mathematica and REDUCE.



Язык: en

Рубрика: Математика/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Издание: 2-nd edition

Год издания: 2006

Количество страниц: 553

Добавлена в каталог: 08.12.2013

Операции: Положить на полку | Скопировать ссылку для форума | Скопировать ID
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Предметный указатель
Theorem, Classification, for Quadrics      404
Theorem, Closure      122ff 190 191 254ff
Theorem, dimension      453
Theorem, Elimination      113 186 386 393 396
Theorem, extension      115 161 166 389
Theorem, Fermat’s last      13
Theorem, Fundamental, of Algebra      4 314
Theorem, Fundamental, of Symmetric Polynomials      312
Theorem, Geometric Extension      121 385 386 395
Theorem, Hilbert Basis      14 30 74ff 78 167 204 205 224 336 393
Theorem, implicit function      286 481
Theorem, intermediate value      423
Theorem, Isomorphism      226 339
Theorem, Lasker — Noether      208 209
Theorem, Molien’s      334 338 520
Theorem, Noether’s      331 336
Theorem, Normal Form for Quadrics      402
Theorem, Pappus’s      299 357 358 427
Theorem, Pascal’s Mystic Hexagon      424 426 427
Theorem, Polynomial Implicitization      126 342
Theorem, Projective Extension      389
Theorem, Pythagorean      289
Theorem, Rational Implicitization      130
Toumier, E.      39 42 44 149 188
Trager, B.      176 206
Transcendence degree      470 471
Transformation, affine      273
Transformation, projective linear      400
Traverso, C.      108 521
Triangular form      304
Twisted cubic tangent surface of      see "Surface tangent"
Unique factorization of polynomials      149
Uniqueness question in invariant theory      327 338 341
van Dam, A.      521
Van der Waerden, B.      154
Variety, affine      5
Variety, dual      348
Variety, irreducible      195 201 203 205 215 235 254 294 342 373 378 462 469
Variety, irreducible component of      293 415 462 480
Variety, linear      9 363
Variety, minimum principle      260
Variety, of an ideal (V(I))      77 372
Variety, projective      363
Variety, rational      250 252
Variety, reducible      215
Variety, Segre      384
Variety, subvariety of      236
Variety, unirational      17
Variety, zero-dimensional      233
Vasconcelos, W.      176 206
Walker, R.      419 423 424
Wang, D.      309 510 520
Warren, J.      154 520 521
Weight order      see "Monomial ordering"
Weights      396
Weispfenning, V.      80 108 176 186 206 233 278 517 522
Well-ordering      53 54 70
Whitney umbrella      see "Surface Whitney
Wiles, A.      13
Winkler, F.      109
Wolfram, S.      511
Wu, W.-T.      302 309 520
Wu’s Method      302ff 309 520
Zacharias, G.      176 206
Zariski, dense set      468
Zelevinsky, A.      154
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