Ideals, Varieties, and Algorithms: An Introduction to Computational Algebraic Geometry and Commutative Algebra, Volume 10

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Springer Science & Business Media, Feb 14, 2007 - Computers - 551 pages
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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. Although the algorithmic roots of algebraic geometry are old, it is only in the last forty years that computational methods have regained their earlier prominence. New algorithms, coupled with the power of fast computers, have led to both theoretical advances and interesting applications, for example in robotics and in geometric theorem proving.

In addition to enhancing the text of the second edition, with over 200 pages reflecting changes to enhance clarity and correctness, this third edition of Ideals, Varieties and Algorithms includes:

A significantly updated section on Maple in Appendix C

Updated information on AXIOM, CoCoA, Macaulay 2, Magma, Mathematica and SINGULAR

A shorter proof of the Extension Theorem presented in Section 6 of Chapter 3

From the 2nd Edition:

"I consider the book to be wonderful. ... The exposition is very clear, there are many helpful pictures, and there are a great many instructive exercises, some quite challenging ... offers the heart and soul of modern commutative and algebraic geometry." -The American Mathematical Monthly

  

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Contents

IV
1
V
5
VI
14
VII
29
VIII
38
IX
49
X
54
XI
61
XL
279
XLI
291
XLII
307
XLIII
317
XLIV
327
XLV
336
XLVI
345
XLVII
357

XII
69
XIII
75
XIV
82
XV
88
XVI
95
XVII
102
XVIII
115
XIX
123
XX
128
XXI
137
XXII
150
XXIII
162
XXIV
169
XXV
175
XXVI
183
XXVII
193
XXVIII
198
XXIX
204
XXX
210
XXXI
214
XXXII
215
XXXIII
221
XXXIV
230
XXXV
239
XXXVI
248
XXXVII
258
XXXVIII
265
XXXIX
271
XLVIII
368
XLIX
379
L
386
LI
393
LII
408
LIII
422
LIV
439
LV
443
LVI
456
LVII
468
LVIII
477
LIX
484
LX
495
LXI
509
LXII
510
LXIII
511
LXIV
513
LXV
514
LXVII
515
LXVIII
517
LXIX
520
LXX
522
LXXI
524
LXXII
528
LXXIII
530
LXXIV
535
LXXV
541
Copyright

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Page 537 - KLEIN, Felix. Lectures on the Ikosahedron, and the Solution of Equations of the Fifth Degree. Translated by GG MORRICE.

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About the author (2007)

David W. Cox is a graduate of Wofford College and Southern Baptist Theological Seminary and holds the M.Div. and doctor of ministry degrees. He completed a two-year clinical residency as a hospital chaplain and had a ten-year career in emergency medical service. In 1997 Dr. Cox founded the support group S.O.S (Survivors of Suicide). An ordained minister, he is currently in private practice as a professional Christian counselor. He is a survivor of a suicide attempt. He and his family live in Spartanburg, South Carolina.

John Little is one of the world's leading authorities on helping athletes become bigger, stronger, and faster. He lives in Bracebridge, Ontario, Canada.

Donal O'Shea is a professor of mathematics and the dean of faculty and vice president for academic affairs of Mount Holyoke College in Massachusetts. This is his first book for a general audience. He lives in South Hadley, Massachusetts.

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