On the intersections of polynomials and the Cayley–Bacharach theorem

Let R=K[x1,…,xn] and let f1,…,fn be products of linear forms with fi of degree di. Assume that the fi have d1,…,dn 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 fi are arbitrary forms of degree di with the right number of common zeros.