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Harnack's Theorems

Harnack's first theorem states that a real irreducible curve of order $n$ cannot have more than

{\textstyle{1\over 2}}(n-1)(n-2)-\sum s_i(s_i-1)+1

circuits (Coolidge 1959, p. 57).

Harnack's second theorem states that there exists a curve of every order with the maximum number of circuits compatible with that order and with a certain number of double points, provided that number is not permissible for a curve of lower order (Coolidge 1959, p. 61).


Coolidge, J. L. A Treatise on Algebraic Plane Curves. New York: Dover, 1959.

© 1996-9 Eric W. Weisstein