Invariant theoretic characterization of subdiscriminants of matrices

An invariant theoretic characterization of subdiscriminants of matrices is given. The structure as a module over the special orthogonal group of the minimal degree non-zero homogeneous component of the vanishing ideal of the variety of real symmetric matrices with a bounded number of different eigenvalues is investigated. These results are applied to the study of sum of squares presentations of subdisciminants of real symmetric matrices.