Solving Multiparameter Eigenvalue problem I have a multiparameter polynomial eigenvalue problem of the form:
\begin{equation}
(\alpha^2 A + \alpha\beta B + \alpha\gamma C + \alpha D + \beta^2 E + \beta\gamma F + \beta G + \gamma^2 H + \gamma J + K)\cdot{t} = 0
\end{equation}
where $\alpha$, $\beta$, and $\gamma$ are unknown scalars, and A...K are 4x4 matrices.
We can linearize it to:
\begin{equation}
\left( \alpha
\begin{bmatrix}
A & B & C & D \\
0 & 0 & 0 & I \\
0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 \\
\end{bmatrix}
+ \beta
\begin{bmatrix}
0 & E & F & G \\
0 & 0 & 0 & 0 \\
0 & 0 & 0 & I \\
0 & 0 & 0 & 0 \\
\end{bmatrix}
+ \gamma
\begin{bmatrix}
0 & 0 & H & J \\
0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 \\
0 & 0 & 0 & I \\
\end{bmatrix}
+
\begin{bmatrix}
0 & 0 & 0 & K \\
-I & 0 & 0 & 0 \\
0 & -I & 0 & 0 \\
0 & 0 & -I & 0 \\
\end{bmatrix} \right) 
\begin{bmatrix}
\alpha t \\
\beta t \\
\gamma t \\
t
\end{bmatrix}
= 0,
\end{equation}
which we will rewrite as,
\begin{equation}
(\alpha X + \beta Y + \gamma Z + W) \hat{t} = 0
\end{equation}
where $\hat{t} = [\alpha t^\top \; \beta t^\top \; \gamma t^\top \; t^\top]^\top $. This is a multiparameter eigenvalue problem. However, I have not been able to find literature for methods to readily solve this (there is plenty of literature for 2-parameter, but I cannot find > 2 parameters). 
Could anybody point me to relevant methods or papers that could explain how to solve this?

Could you be more precise about what you want to compute? The solutions $(\alpha,\beta,\gamma)$ and $t$ to the first equation do not form a countable set, since you can for instance fix (arbitrary) $\beta$ and $\gamma$ and get $2n$ solution pairs $(\alpha,t)$. 
