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Noether's Fundamental Theorem

If two curves $\phi$ and $\psi$ of Multiplicities $r_i\not=0$ and $s_i\not=0$ have only ordinary points or ordinary singular points and Cusps in common, then every curve which has at least Multiplicity


at every point (distinct or infinitely near) can be written

f\equiv \phi\psi'+\psi\phi'=0,

where the curves $\phi'$ and $\psi'$ have Multiplicities at least $r_i-1$ and $s_i-1$.


Coolidge, J. L. A Treatise on Algebraic Plane Curves. New York: Dover, pp. 29-30, 1959.

© 1996-9 Eric W. Weisstein