Let be a collection of distinct points. Then the number of conditions imposed by on forms of degree is called the Hilbert function of . If curves and of degrees and meet in a collection of points, then for any , the number of conditions imposed by on forms of degree is independent of and and is given by

where the Binomial Coefficient is taken as 0 if (Cayley 1843).

**References**

Eisenbud, D.; Green, M.; and Harris, J. ``Cayley-Bacharach Theorems and Conjectures.'' *Bull. Amer. Math. Soc.*
**33**, 295-324, 1996.

© 1996-9

1999-05-25