Polynomials with and without determinantal representations (Q445815)

The paper is studying the problem of writing real zero (RZ) polynomials as determinants of linear matrix polynomials. More specifically, it is shown that almost no real zero polynomial admits a determinantal representation. In fact, there are dimensional differences between the set of real zero polynomials and the set of polynomials with a determinantal representation. A characterization is given for polynomials of which some power has a determinantal representation. It is proved that any quadratic RZ-polynomial admits a determinantal representation., after taking a high enough power. An explicit method is given for the construction of a determinantal representation.