Monic symmetric/Hermitian Determinantal Representations of Multivariate polynomials

In convex algebraic geometry, a problem that has generated a lot of interest is the problem of determinantal representations of polynomials and these determinantal polynomials play a crucial role in semidefinite programming problems . In my talk, we discuss the problem of representing a multivariate real polynomial of (total) degree d as the determinant of a monic linear matrix polynomial whose coefficient matrices are either symmetric/ Hermitian matrices of order d. In particular, I shall talk about a complete characterization of quadratic determinantal polynomials and propose a method to compute a monic symmetric/Hermitian determinatal representation for a bivariate polynomial if it exists using the theory of majorization and exterior algebra.