Conics and Cubics offers an accessible and well illustrated introduction to algebraic curves. By classifying irreducible cubics over the real numbers and proving that their points form Abelian groups, the book gives readers easy access to the study of elliptic curves. It includes a simple proof of Bezout’s Theorem on the number of intersections of two curves. The subject area is described by means of concrete and accessible examples. The book is a text for a one-semester course.
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Editorial Reviews
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"...This book therefore belongs to the admirable tradition of laying the foundations of a difficult and potentially abstract subject by means of concrete and accessible examples. ... Two major strengths of the book are its historical perspective, in the form of informative introductions to the chapters which give the main developments in non-technical language, and its exercises, which are numerous and interesting." Peter Giblin for MathSciNet
From the reviews of the second edition:
"Algebraic geometry is a hard subject. ... But could it, or at least some of it, be presented, at the undergraduate level? This book attempts to do that. ... At the beginning of each of the four chapters, the author provides a synopsis of the historical development of the subject. And within each section many exercises are provided for further discussion and illumination. ... And the author manages to keep things concrete. So, the end result is a book which is accessible … ." (Donald L. Vestal, MathDL – online, October, 2006)
From the Back Cover
Conics and Cubics is an accessible introduction to algebraic curves. Its focus on curves of degree at most three keeps results tangible and proofs transparent. Theorems follow naturally from high school algebra and two key ideas: homogenous coordinates and intersection multiplicities.
By classifying irreducible cubics over the real numbers and proving that their points form Abelian groups, the book gives readers easy access to the study of elliptic curves. It includes a simple proof of Bezout's Theorem on the number of intersections of two curves.
The book is a text for a one-semester course on algebraic curves for junior-senior mathematics majors. The only prerequisite is first-year calculus.
The new edition introduces the deeper study of curves through parametrization by power series. Two uses of parametrizations are presented: counting multiple intersections of curves and proving the duality of curves and their envelopes.
About the first edition:
"The book...belongs in the admirable tradition of laying the foundations of a difficult and potentially abstract subject by means of concrete and accessible examples."
- Peter Giblin, MathSciNet
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6 of 7 people found the following review helpful:
5.0 out of 5 stars
A perfect introduction!,
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This review is from: Conics and Cubics: A Concrete Introduction to Algebraic Curves (Undergraduate Texts in Mathematics) (Hardcover)
I'm a freshman in college and I wanted to start doing research in mathematics. This was a perfect way for me to get a leg up on my peers as algebraic geometry is a vital tool in mathematics today and most undergraduates never get an exposure to it.
The language is formal enough to be a textbook, but it's certainly not detached, it's much like reading the transcript of a lecture. The only knowledge that the book assumes is single variable calculus and high school algebra and geometry. Overall, the book is just very accessible and well worth the price.
8 of 12 people found the following review helpful:
4.0 out of 5 stars
Welcome Addition to the Literature,
By
This review is from: Conics and Cubics: A Concrete Introduction to Algebraic Curves (Undergraduate Texts in Mathematics) (Hardcover)
This book may be the most elementary introduction to algebraic geometry. Still it is roughly senior level and unlike the review above I am not sure that it is suitable for students in secondary education. Nonetheless, it fills a niche that has been largely vacant in the undergraduate literature and I recommend it to serious students and undergraduates alike.
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Inside This Book
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Key Phrases - Statistically Improbable Phrases (SIPs):
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reduced parametrizations, polarity interchanges, plex projective plane, dean plane, complex affine plane, hirational transformations, parametrizing curves, real projective plane, equivalent parametrizations, intersection multiplicities, nonzero constant term, third intersection, branch tangent, transformation that maps, unique conic, intersection multiplicity, transformation fixes, cubic can, nonsingular complex, counting multiplicities, consider the following result, hexagon abcdef, irreducible curve, basic polarity, irreducible cubics Key Phrases - Capitalized Phrases (CAPs): (learn more)
Bezout's Theorem, Pascal's Theorem, Theorem Let, Fundamental Theorem of Algebra, Proof Let, Brianchon's Theorem, Miquel's Theorem, Proof There, Weierstrass P-function, Involution Theorem, Max Noether, Proof Assume, Proof By Theorem New!
Concordance | Text Stats Browse Sample Pages:
Front Cover | Table of Contents | First Pages | Index | Back Cover | Surprise Me!
reduced parametrizations, polarity interchanges, plex projective plane, dean plane, complex affine plane, hirational transformations, parametrizing curves, real projective plane, equivalent parametrizations, intersection multiplicities, nonzero constant term, third intersection, branch tangent, transformation that maps, unique conic, intersection multiplicity, transformation fixes, cubic can, nonsingular complex, counting multiplicities, consider the following result, hexagon abcdef, irreducible curve, basic polarity, irreducible cubics Key Phrases - Capitalized Phrases (CAPs): (learn more)
Bezout's Theorem, Pascal's Theorem, Theorem Let, Fundamental Theorem of Algebra, Proof Let, Brianchon's Theorem, Miquel's Theorem, Proof There, Weierstrass P-function, Involution Theorem, Max Noether, Proof Assume, Proof By Theorem New!
Concordance | Text Stats Browse Sample Pages:
Front Cover | Table of Contents | First Pages | Index | Back Cover | Surprise Me!
Citations (learn more)
70
books
cite this book:
See all 70 books citing this book
- Conics and Cubics: A Concrete Introduction to Algebraic Curves (Undergraduate Texts in Mathematics) by Robert Bix in Front Matter (1), and Front Matter (2)
- Introduction to Cryptography (Undergraduate Texts in Mathematics) by Johannes Buchmann in Front Matter
- Differential Equations: An Introduction with Mathematica® (Undergraduate Texts in Mathematics) by Clay C. Ross in Front Matter
- Introduction to the Mathematics of Finance: From Risk Management to Options Pricing (Undergraduate Texts in Mathematics) by Steven Roman in Front Matter
- The Heritage of Thales (Undergraduate Texts in Mathematics / Readings in Mathematics) by W.S. Anglin in Back Matter
See all 70 books citing this book
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