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Several questions in complex analysis lead naturally to the study of bihomogeneous polynomials r(z, z) on Cn×Cn for which r(z, z) ‖z‖ = ‖h(z)‖ for some natural number d and a holomorphic polynomial mapping h = (h1, . . . , hK) from Cn to CK . 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[z1, . . . , zn] and apply a well-known result of Macaulay to estimate some natural quantities.

An application of Macaulay’s estimate to sums of squares problems in several complex variables