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In a detailed and comprehensive introduction to the theory of plane algebraic curves, the authors examine this classical area of mathematics that both figured prominently in ancient Greek studies and remains a source of inspiration and a topic of research to this day. Arising from notes for a course given at the University of Bonn in Germany, "Plane Algebraic Curves" reflects the authors concern for the student audience through its emphasis on motivation, development of imagination, and understanding of basic ideas. As classical objects, curves may be viewed from many angles. This text also provides a foundation for the comprehension and exploration of modern work on singularities.
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In the first chapter one finds many special curves with very attractive geometric presentations - the wealth of illustrations is a distinctive characteristic of this book - and an introduction to projective geometry (over the complex numbers). In the second chapter one finds a very simple proof of Bezout's theorem and a detailed discussion of cubics. The heart of this book - and how else could it be with the first author - is the chapter on the resolution of singularities (always over the complex numbers). (...) Especially remarkable is the outlook to further work on the topics discussed, with numerous references to the literature. Many examples round off this successful representation of a classical and yet still very much alive subject.
(Mathematical Reviews)
Inhaltsverzeichnis
I. History of algebraic curves.- 1. Origin and generation of curves.- 2. Synthetic and analytic geometry.- 3. The development of projective geometry.- II. Investigation of curves by elementary algebraic methods.- 4. Polynomials.- 5. Definition and elementary properties of plane algebraic curves.- 6. The intersection of plane curves.- 7. Some simple types of curves.- III. Investigation of curves by resolution of singularities.- 8. Local investigations.- 9. Global investigations.- Bibliography.- Index.
Über den Autor / die Autorin
Egbert Brieskorn was a Professor of Mathematics at the University of Bonn, Germany.
Horst Knörrer is a Professor of Mathematics at the ETH Zurich, Switzerland.
Zusammenfassung
In a detailed and comprehensive introduction to the theory of plane algebraic curves, the authors examine this classical area of mathematics that both figured prominently in ancient Greek studies and remains a source of inspiration and a topic of research to this day. Arising from notes for a course given at the University of Bonn in Germany, “Plane Algebraic Curves” reflects the authorsʼ concern for the student audience through its emphasis on motivation, development of imagination, and understanding of basic ideas. As classical objects, curves may be viewed from many angles. This text also provides a foundation for the comprehension and exploration of modern work on singularities.
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In the first chapter one finds many special curves with very attractive geometric presentations ‒ the wealth of illustrations is a distinctive characteristic of this book ‒ and an introduction to projective geometry (over the complex numbers). In the second chapter one finds a very simple proof of Bezout’s theorem and a detailed discussion of cubics. The heart of this book ‒ and how else could it be with the first author ‒ is the chapter on the resolution of singularities (always over the complex numbers). (…) Especially remarkable is the outlook to further work on the topics discussed, with numerous references to the literature. Many examples round off this successful representation of a classical and yet still very much alive subject.
(Mathematical Reviews)
Zusatztext
“It provides a comprehensive overview for all who are interested in GO with an emphasis on theory and algorithms.” (W. Huyer, Monatshefte für Mathematik, 2015)
“This is a masterly expositional work in which the conversational style of narrative never leaves the reader in doubt about the direction of enquiry. … the richness of this publication really resides in the fascinating range of mathematical ideas that support its main line of enquiry. … it can be read selectively at so many different levels up to the postgraduate stage.” (PeterRuane,The Mathematical Association of America, January, 2013)
Bericht
"It provides a comprehensive overview for all who are interested in GO with an emphasis on theory and algorithms." (W. Huyer, Monatshefte für Mathematik, 2015)
"This is a masterly expositional work in which the conversational style of narrative never leaves the reader in doubt about the direction of enquiry. ... the richness of this publication really resides in the fascinating range of mathematical ideas that support its main line of enquiry. ... it can be read selectively at so many different levels up to the postgraduate stage." (PeterRuane,The Mathematical Association of America, January, 2013)
