The First Part of Hilbert's 16th Problem
In 1876 Harnack investigated algebraic curves and found that curves of degree n could have no more than
separate components in the real projective plane. Furthermore he showed how to construct curves that attained that upper bound, and thus that it was the best possible bound. Curves with that number of components are called M-curves.
Hilbert had investigated the M-curves of degree 6, and found that the 11 components always were grouped in a certain way. His challenge to the mathematical community now was to completely investigate the possible configurations of the components of the M-curves.
Furthermore he requested a generalization of Harnack's Theorem to algebraic surfaces and a similar investigation of surfaces with the maximum number of components.
Read more about this topic: Hilbert's Sixteenth Problem
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