Invariant theoretic characterization of subdiscriminants of matrices

TL;DR

This paper provides an invariant theoretic framework for understanding subdiscriminants of matrices, exploring their algebraic structure and applications to sum of squares representations.

Contribution

It introduces a new invariant theoretic characterization of subdiscriminants and analyzes their module structure over the special orthogonal group.

Findings

Characterization of subdiscriminants as modules over SO(n)

Identification of minimal degree components of the vanishing ideal

Application to sum of squares representations of subdiscriminants

Abstract

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.