  Let $p$ be a real zero polynomial in $n$ variables. Then $p$ defines a
rigidly convex set $C(p)$. We construct a linear matrix inequality of size
$n+1$ in the same $n$ variables that depends only on the cubic part of $p$ and
defines a spectrahedron $S(p)$ containing $C(p)$. The proof of the containment
uses the characterization of real zero polynomials in two variables by Helton
and Vinnikov. We exhibit many cases where $C(p)=S(p)$.
  In terms of optimization theory, we introduce a small semidefinite relaxation
of a potentially huge hyperbolic program. If the hyperbolic program is a linear
program, we introduce even a finitely convergent hierachy of semidefinite
relaxations. With some extra work, we discuss the homogeneous setup where real
zero polynomials correspond to homogeneous polynomials and rigidly convex sets
correspond to hyperbolicity cones.
  The main aim of our construction is to attack the generalized Lax conjecture
saying that $C(p)$ is always a spectrahedron. We show that the ``weak real zero
amalgamation conjecture'' of Sawall and the author would imply the following
partial result towards the generalized Lax conjecture: Given finitely many
planes in $\mathbb R^n$, there is a spectrahedron containing $C(p)$ that
coincides with $C(p)$ on each of these planes. This uses again the result of
Helton and Vinnikov.
