Real Algebraic and Analytic Geometry |

Polynomials with and without determinantal representations.

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Submission: 2010, August 11.

*Abstract:
The problem of writing real zero polynomials as determinants of linear matrix polynomials has recently
attracted a lot of attention. Helton and Vinnikov \cite{hevi} have proved that any real zero polynomial in two variables
has a determinantal representation. Br\"and\'{e}n \cite{bran} has shown that the result does not extend to arbitrary numbers
of variables, disproving the generalized Lax conjecture. We provide a large class of surprisingly simple real zero
polynomials that do not have a determinantal representation, improving upon Br\"and\'{e}n's result.
We 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.*

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