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A Guide to Plane Algebraic Curves (Dolciani Mathematical Expositions), by Keith Kendig
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This Guide is a friendly introduction to plane algebraic curves. It emphasizes geometry and intuition, and the presentation is kept concrete. You'll find an abundance of pictures and examples to help develop your intuition about the subject, which is so basic to understanding and asking fruitful questions. Highlights of the elementary theory are covered, which for some could be an end in itself, and for others an invitation to investigate further. Proofs, when given, are mostly sketched, some in more detail, but typically with less. References to texts that provide further discussion are often included.
Computer algebra software has made getting around in algebraic geometry much easier. Algebraic curves and geometry are now being applied to areas such as cryptography, complexity and coding theory, robotics, biological networks, and coupled dynamical systems. Algebraic curves were used in Andrew Wiles' proof of Fermat's Last Theorem, and to understand string theory, you need to know some algebraic geometry. There are other areas on the horizon for which the concepts and tools of algebraic curves and geometry hold tantalizing promise. This introduction to algebraic curves will be appropriate for a wide segment of scientists and engineers wanting an entrance to this burgeoning subject.
- Sales Rank: #1535779 in Books
- Published on: 2011-07-27
- Original language: English
- Number of items: 1
- Dimensions: 8.98" h x .43" w x 5.98" l, .45 pounds
- Binding: Hardcover
- 193 pages
Review
Algebraic curves have regained a prominent position in mathematics. In light of their importance, the goal of this book is to provide a reasonable understanding of algebraic curves and their use. Beginning with standard curves (polynomial, parametric, conic, and user defined), Kendig expands the study by first shrinking the plane to a disk by adjoining points at infinity, and then shifting the domain from real to complex numbers to establish Bezout's theorem. Given this context, the study shifts further to determining the topological properties of algebraic curves, relating genus to a polynomial's degree, investigating singularities, and using compact Riemann surfaces. Throughout, the author emphasizes the geometry and intuitive aspects of algebraic curves, without delving into a tedious chain of proofs. He briefly considers applications of algebraic curves, ranging from Andrew Wiles's special use of elliptic curves in his proof of Fermat's last theorem to their use in cryptography, dynamical systems, and robotics. Readers should be familiar with basic ideas from geometric topology, complex analysis, and abstract algebra. Since Kendig developed the content as a guide rather than a textbook, no problem sets are included, but the author does suggest appropriate textbooks in a bibliography. --J. Johnson, CHOICE Magazine
This book is a straightforward and simple introduction to plane algebraic curves and would be of interest to anyone wanting a good overview of the subject. The material could also serve as a useful supplement for students taking an introductory course on algebraic curves or for mathematicians who would like to learn about the subject. Even though the book is written as a guide, readers will need some basic understanding of complex analysis, field theory, and topology to comprehend the subject matter completely.
The book is informal in its approach and focuses on building intuition through a variety of pictures and clarifying examples without becoming mired in detailed proofs and exercises. Chapter 1 explores algebraic curves in the real plane; in subsequent chapters, the canvas expands to include the complex numbers. The book's final chapters focus more on the geometric properties of algebraic curves and conclude with a foray into the topic of Riemann surfaces.
A Guide to Plane Algebraic Curves is an accessible and well-written book that anyone with an interest in this beautiful subject will surely appreciate and find useful. --Marc Michael, Mathematics Teacher
About the Author
Keith Kendig received his Ph.D. from UCLA in Algebraic Geometry, and subsequently spent two years at the Institute for Advanced Study. He is the author of Elementary Algebraic Geometry, and two other books, Conics and Sink or Float? Thought Problems in Math and Physics. In 2000, he received the Lester Ford Prize for his Monthly article, Is a 2000-Year-Old Formula Still Keeping Some Secrets? He is currently an associate editor of Mathematics Magazine and is also on the editorial board of the Spectrum Series of the MAA.
Most helpful customer reviews
6 of 6 people found the following review helpful.
An absolutely great book intended for curious undergraduates or non-experts
By slizard
An absolutely great book intended for curious undergraduates or non-experts. Assumes basically just a solid knowledge of calculus and yet manages to take you far, in a very natural way! Focuses on main ideas which are also tied with other areas of math (e.g topology or computer visualization). Very entertaining while not simplistic at all. A true gem.
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