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

Abstract:

Several questions in complex analysis lead naturally to the study of bihomogeneous polynomials $r(z,\bar {z})$ on $\mathbb {C}^n \times \mathbb {C}^n$ for which $r(z,\bar {z})\left \lVert z \right \rVert ^{2d}=\left \lVert h(z) \right \rVert ^2$ for some natural number $d$ and a holomorphic polynomial mapping $h=(h_1, \ldots , h_K)$ from $\mathbb {C}^n$ to $\mathbb {C}^K$. 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 $\mathbb {C}[z_1,\ldots ,z_n]$ and apply a well-known result of Macaulay to estimate some natural quantities.