Real Algebraic and Analytic Geometry
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314. Tim Netzer, Andreas Thom:
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.

Full text, 20p.: dvi 132k, pdf 363k.