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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. [ABSTRACT FROM AUTHOR]

The product of a Hermitian matrix and a positive semidefinite matrix has only real eigenvalues. We present bounds for sums of eigenvalues of such a product. [ABSTRACT FROM AUTHOR]

If a convex body C in …n has modular and irreducible face lattice and C is not strictly convex, there is a face-preserving homeomorphism from C to a set of positive-semidefinite Hermitian matrices of trace 1 over …, … or …, or C has dimension 8, 14 or 26. [ABSTRACT FROM AUTHOR]