On the intersections of polynomials and the Cayley–Bacharach theorem

Abstract

Let R= K[ x 1,…, x n ] and let f 1,…, f n be products of linear forms with f i of degree d i . Assume that the f i have d 1,…, 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.