Despite the title, this is probably actually a question in linear algebra or algebraic geometry. Let me write the question(s) first, before I explain the background. 


**Problems**


Let $h^{\mu\nu}\_{ij}$ represent a map from $\mathbb{R}^4\otimes\mathbb{R}^4$ to $\mathbb{R}^2\otimes\mathbb{R}^2$ (here $\mu\nu$ are indices in the $\mathbb{R}^4$ directions, and $ij$ are in $\mathbb{R}^2$ directions). We shall assume that $h$ is symmetric swapping the $\mu,\nu$ indices and also symmetric swapping the $i,j$ indices. Then for any $\xi\_\mu$ in $\mathbb{R}^4$, the object $H(\xi):= h^{\mu\nu}\_{ij}\xi\_\mu\xi\_\nu$ is a symmetric bilinear form on $\mathbb{R}^2$. We say that $\xi$ is characteristic if $H(\xi)$ is degenerate. In other words, $\xi$ is characteristic if $\det(H(\xi)) = 0$. 


Since $H(\xi)$ is quadratic in $\xi$, the determinant is an 8th degree homogeneous polynomial in $\xi$. Furthermore, by definition if $\xi$ is characteristic, so is $-\xi$. Observe also that in general the characteristic set will have multiple sheets. 


*Question 1, very specific*


Does there exist an $h$ such that the characteristic surface is given by $\xi\_1^4 + \xi\_2^4 + \xi\_3^4 - \xi\_4^4 = 0$? 


*Question 2, slightly more general*


In general are there any obstructions to having a sheet of the characteristic surface described by the zero set of an irreducible (over the reals) polynomial of degree strictly higher than 2?


*Question 3, even more general*


What if we relax the condition on $h$ so that it is a map from $\mathbb{R}^m\otimes\mathbb{R}^m$ to $\mathbb{R}^d\otimes\mathbb{R}^d$ with the same symmetric properties. Define $H(\xi)$ analogously. Can a sheet of the characteristic surface have algebraic degree more than 2?


I'm particularly interested in concrete examples. 


**Motivation**


This comes from the study of hyperbolic systems partial differential equations. Recall that a second degree partial differential equation 
$$ h^{\mu\nu}\_{ij} \partial\_\mu\partial\_\nu u^i = 0 $$
is said to be strictly hyperbolic in the direction of $e\_\mu$ if the characteristic polynomial (a polynomial in $t$) $\det(H(x\_\mu - te\_\mu))$ is hyperbolic for any fixed $x\_\mu$ linearly independent from $e\_\mu$ and that the roots are distinct (it is enough that the second condition only holds for all by finitely many $x\_\mu$ modulo $e\_\mu$). 


The classical examples for strictly hyperbolic systems (wave equation, crystal optics, etc) all have the sheets of the characteristic surfaces being linearly transformed versions of the standard quadratic double cone: in other words there exists a basis of $\mathbb{R}^m$ such that a sheet is given by $\sum\_{i = 1}^{m-1} e\_i^2 - e\_m^2 = 0$. 


I am guessing that for strictly hyperbolic systems in fact all sheets must be of this form due to homogeneity (though please let me know if I am wrong). 


So my question is: is it possible for a non-strictly hyperbolic system (but one still hyperbolic) where some of the sheets have higher multiplicity to not come from "the square of a quadratic sheet" but from a genuinely quartic or higher polynomial? 


**Postscript**


Please do let me know if you need any clarification on my question. Thanks.


**Update**


I struck out question 1 for the following reason: in view of my motivation from hyperbolic polynomials arising from second order PDEs, the answer is negative. The argument is thus: for a hyperbolic system of PDEs, the time-like direction $\xi\_4$ should have its corresponding $h^{44}\_{ij}$ negative definite, whereas the space-like directions $\xi\_1,\xi\_2,\xi\_3$ should have their corresponding $h^{aa}\_{ij}$ positive definite. A simple computation shows that the coefficient to the $\xi\_a^4$ term in $\det H(\xi)$ must be $\det h^{aa}\_{ij}$. If the target space is two dimensional, both positive definite and negative definite matrices have positive determinants. So for any hyperbolic polynomial arising from a second order system of PDEs, the coefficients for $\xi\_a^4$ must be positive. 



Characteristic surface for systems of PDE
There is a natural example where an irreducible component of the symbol, a hyperbolic polynomial, is of degree higher than $2$. It occurs in compressible magnetohydrodynamics (MHD). It is a coupling of the Euler system of compressible, inviscid gas dynamics, with Maxwell's equations, when the magnetic part dominates the EM field. It consists of $8$ conservation laws, governing mass density ($1$), momentum ($3$), energy ($1$) and magnetic field ($3$). When linearizing around a uniform state, we obtain a linear, first-order hyperbolic system. The determinant of the symbol splits into the square of a linear factor (pure transport), a quadratic factor (Alfvén mode) and an irreducible factor of degree $4$. The roots of the latter are the velocities in direction $\xi$ of the fast/slow backward/forward waves.


You can also get a complex structure in nonlinear models of EM fields. See [here](http://www.umpa.ens-lyon.fr/~serre/DPF/LTT2.pdf).


