Algebraic properties of Hermitian sums of squares, II

Abstract:

We study real bihomogeneous polynomials $r(z,\bar {z})$ in $n$ complex variables for which $r(z,\bar {z}) \left \lVert {z} \right \rVert ^2$ 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 $r(z,\bar {z}) \left \lVert {z} \right \rVert ^2$ 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 $r(z,\bar {z}) \left \lVert {z} \right \rVert ^2$ can be reformulated as questions about polynomial ideals. We take this approach and apply purely algebraic tools to obtain constraints on the signature of $r$.