Polynomials with and without determinantal representations

TL;DR

This paper investigates the limitations of representing real zero polynomials as determinants of linear matrix polynomials, showing that such representations are rare and providing explicit examples and characterizations.

Contribution

It proves that almost no real zero polynomial admits a determinantal representation, establishes bounds on matrix sizes, and characterizes polynomials whose powers have such representations.

Findings

Most real zero polynomials lack determinantal representations.

Quadratic real zero polynomials become representable after taking a sufficiently high power.

Explicit representations are constructed and characterized up to unitary equivalence.

Abstract

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.