Harnack's curve theorem

Harnack's curve theorem

In real algebraic geometry, Harnack's curve theorem states when a curve of degree m can have c components. For any real plane algebraic curve of degree m, the number of components c is bounded by

:frac{1-(-1)^m}{2} le c le frac{(m-1)(m-2)}{2}+1.

Moreover, any number of components in this range of possible values can be attained. A curve which attains the maximum number of real components is called an "M-curve". The Trott curve, a quartic with four components, is an example of an M-curve. The maximum number is one more than the maximum genus of a curve of degree m, attained when the curve is nonsingular. This theorem formed the background to Hilbert's sixteenth problem.

References

*D. A. Gudkov, "The topology of real projective algebraic varieties", Uspekhi Mat. Nauk 29 (1974), 3-79 (Russian), English transl., Russian Math. Surveys 29:4 (1974), 1-79
*C. G. A. Harnack, "Über Vieltheiligkeit der ebenen algebraischen Curven", Math. Ann. 10 (1876), 189-199
*G. Wilson, "Hilbert's sixteenth problem", Topology 17 (1978), 53-74


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