Definite Determinantal Representations of Multivariate Polynomials

In this paper, we consider the problem of representing a multivariate polynomial as the determinant of a definite (monic) symmetric/Hermitian linear matrix polynomial (LMP). Such a polynomial is known as determinantal polynomial. Determinantal polynomials can characterize the feasible sets of semidefinite programming problems that motivates us to deal with this problem. We introduce the notion of generalized mixed discriminant of matrices which translates the determinantal representation problem into computing a point of a real variety of a specified ideal. We develop an algorithm to determine such a determinantal representation of a bivariate polynomial of degree $d$. Then we propose a heuristic method to obtain a monic symmetric determinantal representation of a multivariate polynomial of degree $d$.