On the intersections of polynomials and the Cayley-Bacharach theorem

Let R=K[x₁,.., x_n] and let f₁, ...,f_n be products of linear forms with f_i of degree d_i. Assume that the f_i have d₁,...,d_n common zeros. Then we determine the maximum number of those zeros that a form of degree k can go through without going through all of them. This is a version of a conjecture of Eisenbud, Green, and Harris. We suggest a possible method for using this to explore the case where the f_i are arbitrary forms of degree d_i with the right number of common zeros.