  The problem of writing real zero polynomials as determinants of linear matrix
polynomials has recently attracted a lot of attention. Helton and Vinnikov have
proved that any real zero polynomial in two variables has a determinantal
representation. Br\"and\'en has shown that the result does not extend to
arbitrary numbers of variables, disproving the generalized Lax conjecture. We
prove that in fact almost no real zero polynomial admits a determinantal
representation; there are dimensional differences between the two sets. So the
generalized Lax conjecture fails badly. The result follows from a general upper
bound on the size of linear matrix polynomials. We then provide a large class
of surprisingly simple explicit real zero polynomials that do not have a
determinantal representation, improving upon Br\"and\'en's mostly
unconstructive result. We finally characterize polynomials of which some power
has a determinantal representation, in terms of an algebra with involution
having a finite dimensional representation. We use the characterization to
prove that any quadratic real zero polynomial has a determinantal
representation, after taking a high enough power. Taking powers is thereby
really necessary in general. The representations emerge explicitly, and we
characterize them up to unitary equivalence.
