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

$$r_i+s_i-1$$

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 $s_i-1$.

References

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