Real Algebraic Geometry - Arnold, Vladimir I.; Itenberg, Ilia.; Kharlamov, Viatcheslav; Shustin, Eugenii I.; SpringerLink (Online service) (Berlin, Heidelberg : Springer Berlin Heidelberg : Imprint: Springer, 2013)
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 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).
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.
Keywords: Mathematics; Geometry, algebraic; Geometry; Mathematical physics; Mathematics; Algebraic Geometry; Mathematical Methods in Physics; Geometry; Mathematical Applications in the Physical Sciences