Real Algebraic Geometry

1. Front Matter

   Pages i-ix

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2. Introduction

   • Vladimir I. Arnold

   Pages 1-2

3. Geometry of Conic Sections

   • Vladimir I. Arnold

   Pages 3-18

4. The Physics of Conic Sections and Ellipsoids

   • Vladimir I. Arnold

   Pages 19-31

5. Projective Geometry

   • Vladimir I. Arnold

   Pages 33-53

6. Complex Algebraic Curves

   • Vladimir I. Arnold

   Pages 55-75

7. A Problem for School Pupils

   • Vladimir I. Arnold

   Pages 77-83

8. Back Matter

   Pages 85-107

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This book is concerned with one of the most fundamental questions of mathematics: the relationship between algebraic formulas and geometric images.

At one of the first international mathematical congresses (in Paris in 1900), Hilbert stated a special case of this question in the form of his 16^th problem (from his list of 23 problems left over from the nineteenth century as a legacy for the twentieth century).

In spite of the simplicity and importance of this problem (including its numerous applications), it remains unsolved to this day (although, as you will now see, many remarkable results have been discovered).

• algebraic curves
• conic sections
• projective geometry

From the reviews:

“As Arnold’s own 1971 contribution altered the direction of research on this question, this book constitutes a summary of the story coming directly from the master. Summing Up: Recommended. Upper-division undergraduates through researchers/faculty.” (D. V. Feldman, Choice, Vol. 51 (6), February, 2014)

“This book is a translation of lecture notes from Russian with additional comments and notes by the editors. The book is aimed at advanced high school students … . It introduces very advanced topics in a very relaxed and informal style. The book includes some exposition and definitions, some theorems and proofs and a lot of problems with hints or solutions. There are many illustrations to lead the reader to an intuitive understanding of the concepts being developed.” (Thomas C. Craven, Mathematical Reviews, December, 2013)

“It was designed as a set of lecture notes addressed to mathematically talented high-school students, with the main focus on a panoramic view toward elementary, problems concerning geometric objects that can be described by (mostly real) algebraic equations. … this is a highly unusual book on real algebraic curves and various related topics. … a truly irresistible invitation to mathematics in general.” (Werner Kleinert, zbMATH, Vol. 1268, 2013)

• Dept. of Mathematics, Université Pierre et Marie Curie, Paris, France

  Ilia Itenberg

• CNRS - IRMA, University of Strasbourg, Strasbourg, France

  Viatcheslav Kharlamov

• Fac. Exact Sciences, School of Mathematical Sciences, University of Tel Aviv, Tel Aviv, Israel

  Eugenii I. Shustin

• Steklov Mathematical Institute, Russian Academy of Sciences, Moscow, Russia

  Vladimir I. Arnold

Vladimir Arnold is one of the great mathematical scientists of our time. He is famous for both the breadth and the depth of his work.

His first mathematical work, which he did being a third-year student, was the solution of the 13th Hilbert problem about superpositions of continuous functions. His early work on KAM (Kolmogorov, Arnold, Moser) theory solved some of the outstanding problems of mechanics that grew out of fundamental questions raised by Poincare and Birkhoff based on the discovery of complex motions in celestial mechanics. In particular, the discovery of invariant tori, their dynamical implications, and attendant resonance phenomena is regarded today as one of the deepest and most significant achievements in the mathematical sciences.

Arnold has been the advisor to more than 60 PhD students, and is famous for his seminar which thrived on his ability to discover new and beautiful problems. He is known all over the world for his textbooks which include the classics Mathematical Methods of Classical Mechanics, and Ordinary Differential Equations, as well as the more recent Topological Methods m Hydrodynamics written together with Boris Khesin, and Lectures on Partial Differential Equations.

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