Plane algebraic curves. Transl. from the German by Leslie Kay (Q2724091)

The German original of this outstanding introduction to the theory of plane curves appeared in 1994 under the title ``Ebene algebraische Kurven'' [Vieweg Studium: Aufbaukurs Mathematik 67 (Braunschweig 1994; Zbl 0815.14016)].NEWLINENEWLINENEWLINEThe author, well-known in Germany for his masterly skill in writing excellent, thoroughgoing and inspiring textbooks, had provided another (modern) introduction to plane algebraic curves that came with several unique features. In particular, the entire text had been kept as elementary, intuitive and concrete as possible for the beginner, without using any highly advanced concepts or methods from commutative algebra, algebraic geometry, and complex analysis.NEWLINENEWLINENEWLINEOn the other hand, working exclusively over the field of complex numbers, the interplay between algebraic and analytical aspects had been carefully elaborated, and many of the most important classical results on plane algebraic curves (with the exception of the really non-elementary Riemann-Roch theorem) had been derived by explicit, enlightening, very instructive, thereby yet elementary and easily comprehensible ad-hoc methods of proof. Methodological mastery and excellence of exposition combined with mathematical aesthetics and substantial depth of the contents -- that is what characterizes G. Fischer's introductory text on plane algebraic curves.NEWLINENEWLINENEWLINEThe book under review is the just as excellent English translation of the German original text.NEWLINENEWLINENEWLINEFortunately, the author has taken the opportunity of this English translation to even enhance the well-established text a little bit. First, he has added one more section to chapter 3 (``Tangents and singularities''), which is entitled ``Chebyshev curves''. This section gives a brief account of \textit{D. Pecker}'s recent approach to the geometry of the famous Lissajous curves via Chebyshev polynomials [Compos. Math. 87, 1-4 (1993; Zbl 0783.14013)] and fits perfectly into this chapter on singular curves. The second improvement concerns appendix 3 (``The Implicit function theorem''), where \textit{W. Walter}'s recent short and elegant proof of the implicit function theorem via Banach algebras [Elem. Math. 47, 27-32 (1992; Zbl 0895.42025)] has been utilized to replace the former one presented in the German original text.NEWLINENEWLINENEWLINEAssuming only standard undergraduate mathematical concepts from algebra, complex function theory, and elementary topology, this booklet is and remains an excellent first introduction to the realm of algebraic geometry, stimulating for beginners, and an extremely useful source for teachers in this field.NEWLINENEWLINENEWLINEIt is very gratifying to see that this primer of algebraic geometry has been made accessible for students and instructors world-wide.