Algebraic properties of Hermitian sums of squares, II

Abstract

We study real bihomogeneous polynomials r ( z , z ¯ ) r(z,\bar {z}) in n n complex variables for which r ( z , z ¯ ) ‖ z ‖ 2 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 , z ¯ ) ‖ z ‖ 2 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 , z ¯ ) r(z,\bar {z}) and the rank of r ( z , z ¯ ) ‖ z ‖ 2 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 r .