On the Topology of Real Plane Algebraic Curves

Abstract

As early as 1876 Harnack' showed that the maximal number of components (maximal connected subsets) of a real algebraic curve of order n in the projective plane is precisely I(n 1) (n 2) + 1. At the same time Harnack proposed a process for the construction of curves with this maximal number of components. Such curves we shall call in the sequel, M-curves. Harnack showed that these M-curves have no singular points. Take a sphere in the three-dimensional space in which the projective plane containing our algebraic curve is situated, and join the centre of this sphere to every point of the projective plane by a straight line. We thus project the plane on the sphere. A component of algebraic curve is called (or component) if its projection on the sphere consists of two ordinary closed curves. If this projection on the sphere S consists of a single closed curve the corresponding component is called odd. Algebraic curves having no real2 singular points possess at most one odd component. Hence every algebraic curve (having no real singular points) of even order consists of ovals only while a curve of odd order has (besides ovals) exactly one odd component. In 1891 D. Hilbert3 proposed a new method of constructing M-curves. In the same work Hilbert announced without proof that M-curve of order 6 cannot have all its ovals lying outside each other. At least one of these ovals must lie within another oval. Here the words an lies within another oval mean that the cone projecting the first on S lies within the projecting cone of the second oval. Hilbert considers this a remarkable fact, since it proves that M-curves cannot have a too simple topological structure. In his report to the International Mathematical Congress in 1900 on modern problems of mathematics Hilbert considers the investigation of the topology of M-curves and of the corresponding algebraic surfaces as most timely.4 After a series of attempts the above mentioned theorem announced by Hilbert was at last proved in 1911 by K. Rohn.5 In the same work Rohn proved that