Real algebraic geometry. Translated from the Russian by Gerald G. Gould and David Kramer.

Abstract

The author (1937--2010) was one of the most eminent mathematicians of the past fifty years until his untimely death three years ago. His extensive and influential works in dynamical systems, singularity theory, catastrophe theory, real and complex algebraic geometry, topology, classical and celestial mechanics, mathematical physics, and in many other fields displayed an incomparable breadth of mathematical and physical interests as well as an incredible degree of enthusiasm, creativity and mathematical insight. Apart from his pioneering impact to contemporary mathematical research, the author's most precious heritage is the wealth of his brilliant expository writings, including numerous survey articles, textbooks, lecture notes, and other introductory texts. The booklet under review is the English edition of the author's last popular lectures, the Russian original of which was published in 2009, shortly before his death, under the title ``Veshchestvennaya algebraicheskaya geometriya'' by the Moscow Center for Continuous Mathematical Education (MCCE). As the editors point out in their foreword to this English translation, the present book is not intended as a systematic introduction to real algebraic geometry. Actually, 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. As such, the book is written in the author's masterly expository style, interlarded with vivid explanations, numerous illustrating examples, various digressions to other topics in mathematics and physics, historical remarks and beautiful, very instructive pictures. The author's general didactic approach is to present concrete problems whose formulations are accessible to non-specialists in algebraic geometry, on the one hand, and to discuss then their (often stunning) solutions, together with the necessary conceptual framework, on the other hand. As the text is written in a rather informal style, which is very typical for many of the author's expository writings, the editors of the current English translation have provided additional comments at several places in the text. As for the precise contents, the book contains six chapters and an appendix, Chapter 1 serves as a brief introduction to what follows in the sequel. The emphasis is on algebraic curves in the Euclidean plane, that is, on plane conics and their topological properties. Chapter 2 discusses various problems concerning the geometry of conic sections in the plane, whereas Chapter 3 turns to problems related to applications of conic sections and ellipsoids in celestial mechanics and classical gravitation theory. Chapter 4 explains the idea of projective spaces and projective plane curves, together with some related topological aspects, Hilbert's 16th problem, D. A. Gudkov's contributions from the late 1960s towards the arrangement of ovals of certain plane sextics, Harnack's theorem, and the topological classification of smooth real functions on \bbfR^2 with given number of critical points. Chapter 5 is devoted to problems on algebraic curves in the complex plane, both affine and projective, thereby touching upon the degree-genus formula by Riemann-Hurwitz, Riemann surfaces, abelian integrals, and further related topological aspects. Chapter 6 is titled ``A problem for school pupils'' and examines the classical question of how many regions are cut out by n lines in the real plane \bbfR^2. After presenting a very instructive partial answer to this problem, the author invites all school pupils to find a general solution to this (still open) elementary problem and to publish it! Appendix A is a translation of the author's article ``Into how many parts do n lines divide the plane?'' [Mat. Prosveshchenie, Ser.3, 12, 95--104 (2008)], which has been added to the main text at his special request. This appendix complements the discussion of Chapter 6 by the author's original paper on that particular topic. Finally, there is a section containing the editors' comments on the so-called Gudkov conjecture that the author mentioned in Chapter 4. In fact, {\it D. A. Gudkov} [Sov. Math., Dokl. 12, 1559--1563 (1971); translation from Dokl. Akad. Nauk SSSR 200, 1269--1272 (1971; Zbl 0242.14007)] once formulated, in the context of Hilbert's conjectures on the arrangements of ovals of certain projective plane sextice, a further conjecture of this kind, and the author himself indicated a whole new direction of research in this field through his seminal paper ``On the arrangements of ovals of real plane algebraic curves, involutions of four-dimensional smooth manifolds, and the arithmetic of integer quadratic forms'' [Funk. Anal. i Prilozh. 5, No. 3, 1--9 (1971)], which was essentially stimulated by the work of Gudkov and his conjecture. At the end of the book, there are thirty numbered notes containing the editors' comments added to the original text. Altogether, this is a highly unusual book on real algebraic curves and various related topics. It fully reflects the author's inimitable style of thinking and writing mathematics, and it certainly should be seen as another jewel of his rich mathematical heritage. -- Besides, the author's great gift to express his ideas in a very inspiring, even gripping way makes this little booklet a truly irresistible invitation to mathematics in general.