7
$\begingroup$

Let $p \in \mathbb R [x,y,z]$ be a homogeneous irreducible polynomial of degree $d$. 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.

Vinnikov 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.

$\endgroup$
5
$\begingroup$

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

http://arxiv.org/abs/1011.6057

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.