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 16th 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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- Verlag: Springer-Verlag GmbH
- Seitenzahl: 100
- Erscheinungstermin: 15. April 2013
- ISBN-13: 9783642362439
- Artikelnr.: 44220095
"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 focuson 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)