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Let $p(x,y,z)$ be a homogeneous irreducible polynomial of degree $d$, with real coefficients. From Dickson in 1920 we know that there exists $A$, $B$ and $C$ such that

$ det(Ax + By + Cz) = c p(x,y,z)$,

where $c$ is some constant.

Vinnokov in 1988 was able to describe all the non-equivalent determinantal representations as points on the Jacobian variety that are not on the exceptional sub variety. The theoretical work in this paper is relatively constructive, but is still a long way from a numerically stable constructive algorithm for $A$, $B$ and $C$.

Given any polynomial $p(x,y,z)$, can one triple $(A, B,C)$ be constructed in a numerically stable way?

Thanks in advance.

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up vote 2 down vote accepted

This is discussed in a recent work of Plaumann, Sturmfels and Vinzant:

http://arxiv.org/abs/1011.6057

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