Several questions in CR geometry lead naturally to the study of bihomogeneous polynomials r(z,z) on C n ×C n for which r(z,z) z 2d = h(z) 2 for some natural number d and a holomorphic polynomial mapping h = (h 1 , . . . , h K ) from C n to C K . When r has this property for some d, one seeks relationships between d, K, and the signature and rank of the coefficient matrix of r. In this paper, we reformulate this basic question as a question about the growth of the Hilbert function of a homogeneous ideal in C[z 1 , . . . , zn] and apply a well-known result of Macaulay to estimate some natural quantities.