Hyperbolic polynomials are real polynomials whose corresponding hypersurfaces have a special topological property. These polynomials appear in many areas of mathematics, including optimization, combinatorics and differential equations. In an important breakthrough in 2007, Helton and Vinnikov showed that every hyperbolic polynomial in three variables can be written as the determinant of a definite symmetric matrix of linear forms. We'll talk about this theorem and possible methods for constructing such determinantal representations. |