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This book addresses the relationship between algebraic formulas and geometric images. In spite of the simplicity and importance of this problem, it remains unsolved to this day although, as the book shows, many remarkable results have been discovered.

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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).

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Publisher´s Foreword.- Editors´ Foreword.- Introduction.- 2 Geometry of Conic Sections.- 3 The Physics of Conic Sections and Ellipsoids.- 4 Projective Geometry.- 5 Complex Algebraic Curves.- 6 A Problem for School Pupils.- A Into How Many Parts do n Lines Divide the Plane?- Editors´ Comments on Gudkov´s Conjecture.- Notes

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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)

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