  We study real bihomogeneous polynomials $r(z,\bar{z})$ in $n$ complex
variables for which $r(z,\bar{z}) \|z\|^2$ is the squared norm of a holomorphic
polynomial mapping. Such polynomials are the focus of the Sum of Squares
Conjecture, which describes the possible ranks for the squared norm
$r(z,\bar{z}) \|z\|^2$ and has important implications for the study of proper
holomorphic mappings between balls in complex Euclidean spaces of different
dimension. Questions about the possible signatures for $r(z,\bar{z})$ and the
rank of $r(z,\bar{z}) \|z\|^2$ can be reformulated as questions about
polynomial ideals. We take this approach and apply purely algebraic tools to
obtain constraints on the signature of $r$.
