Real algebraic and semi-algebraic sets.

*(English)*Zbl 0694.14006
Actualités Mathématiques. Paris: Hermann Éditeurs des Sciences et des Arts. 340 p. FF 238.00 (1990).

This is an elementary introductory book into real algebraic and semi- algebraic sets.

Chapter 1 starts with properties of real polynomials of one variable. Chapter 2 is a leisurely introduction to semi-algebraic sets. It contains the Tarski-Seidenberg theorem, the Łojasiewicz inequality and the triangulation theorem of semi-algebraic sets. - Chapter 3 is on real algebraic sets. - Chapter 4 is devoted to the notion of complexity, notably on Khovanskij’s theorem on estimating the number of roots of a polynomial equation. - Chapter 5 is a lively summary of plane curves. It contains Harnack’s theorem, and the statements of the beautiful results of the Russian school of Arnol’d, Gudkov, Rokhlin, and Kharlamov.

The elementary nature of this book makes it attractive to beginners in the field. At times it tries to do too much by taking too many detours. My main criticism of the book is chapter 3 where the reviewer’s and H. King’s results on “Blowing down” (glueing via regular maps) real algebraic sets (as well as making them projectively closed) have been presented without crediting the original source. We see the same thing in the triangulation theorem of chapter 2 where Łojasiewiczṕaper on triangulating real algebraic sets have not even been listed in the bibliography. The topology of real algebraic sets have been somewhat misunderstood field where incorrect and redundant duplication of existing results are rampart.

Writing a book is a rear opportunity to put things in perspective; this book doesn’t do that.

Chapter 1 starts with properties of real polynomials of one variable. Chapter 2 is a leisurely introduction to semi-algebraic sets. It contains the Tarski-Seidenberg theorem, the Łojasiewicz inequality and the triangulation theorem of semi-algebraic sets. - Chapter 3 is on real algebraic sets. - Chapter 4 is devoted to the notion of complexity, notably on Khovanskij’s theorem on estimating the number of roots of a polynomial equation. - Chapter 5 is a lively summary of plane curves. It contains Harnack’s theorem, and the statements of the beautiful results of the Russian school of Arnol’d, Gudkov, Rokhlin, and Kharlamov.

The elementary nature of this book makes it attractive to beginners in the field. At times it tries to do too much by taking too many detours. My main criticism of the book is chapter 3 where the reviewer’s and H. King’s results on “Blowing down” (glueing via regular maps) real algebraic sets (as well as making them projectively closed) have been presented without crediting the original source. We see the same thing in the triangulation theorem of chapter 2 where Łojasiewiczṕaper on triangulating real algebraic sets have not even been listed in the bibliography. The topology of real algebraic sets have been somewhat misunderstood field where incorrect and redundant duplication of existing results are rampart.

Writing a book is a rear opportunity to put things in perspective; this book doesn’t do that.

Reviewer: S.Akbulut