Determinantal Representations of Nonnegative Polynomials

In this talk, I will present our work on the study of determinantal representations of globally nonnegative polynomials. We ask whether for a given polynomial p that is globally nonnegative, there exists a symmetric matrix M, the entries of which are quadratic forms of degree 2 such that

• p is the determinant of M
• M is positive semidefinite upon evaluation in any given point.

For suitable polynomials p in three variables, we can answer this question in a hands-on fashion: Assuming that such a determinantal representation M exists, it defines a morphism of sheaves. The cokernel of this map can be explicitly described using methods of cohomology. This allows for verification whether M in the assumed form exists and - provided the answer is positive - a reconstruction of M.

We employ these tools to first reprove a (negative) result of Quarez 2015. We then apply them to the Robinson polynomial and prove that it does not admit such a determinantal representation either. This answers a question posed by Buckley and Sivic 2020.