The inequality on the right in 1 is Harnack Inequality. The inequality on the left is part of Corollary 1 of Bézout's theorem (see Section 1.3.B). Thus, Harnack Theorem together with theorems 1.3.B and 1.3.E actually give a complete characterization of the set of topological types of nonsingular plane curves of degree , i.e., they solve problem 1.1.A.
Curves with the maximum number of components (i.e., with components, where is the degree) are called M-curves. Curves of degree which have components are called -curves. We begin the proof of Theorem 1.6.A by establishing that the Harnack Inequality 1.3.B is best possible.
Recall that we obtained a degree 5 M-curve by perturbing the union of two conics and a line . This perturbation can be done using various curves. For what follows it is essential that the auxiliary curve intersect in five points which are outside the two conics. For example, let the auxiliary curve be a union of five lines which satisfies this condition (Figure 6). We let denote this union, and we let denote the M-curve of degree 5 that is obtained using .
We now construct a sequence of auxiliary curves for . We take to be a union of lines which intersect in distinct points lying, for even , in an arbitrary component of the set and for odd in the component of containing .
We construct the M-curve of degree using small perturbation of the union directed to . Suppose that the M-curve of degree has already been constructed, and suppose that intersects transversally in the points of the intersection which lie in the same component of the curve and in the same order as on . It is not hard to see that, for one of the two possible directions of a small perturbation of directed to , the line and the component of that it intersects give components, while the other components of , of which, by assumption, there are
The proof that the left inequality in 1 is best possible, i.e., that there is a curve with the minimum number of components, is much simpler. For example, we can take the curve given by the equation . Its set of real points is obviously empty when is even, and when is odd the set of real points is homeomorphic to (we can get such a homeomorphism onto , for example, by projection from the point .
By choosing the auxiliary curves in different ways in the construction of M-curves in the proof of Theorem 1.6.B, we can obtain curves with any intermediate number of components. However, to complete the proof of Theorem 1.6.A in this way would be rather tedious, even though it would not require any new ideas. We shall instead turn to a less explicit, but simpler and more conceptual method of proof, which is based on objects and phenomena not encountered above.