- 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
:
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 toHilbert'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
Wikimedia Foundation. 2010.