  Several questions in CR geometry lead naturally to the study of bihomogeneous
polynomials $r(z,\bar{z})$ on $\C^n \times \C^n$ for which
$r(z,\bar{z})\norm{z}^{2d}=\norm{h(z)}^2$ 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 $\C[z_1,...,z_n]$ and apply a well-known
result of Macaulay to estimate some natural quantities.
