intuition behind the revelance of hyperbolic polynomial A hyperbolic polynomial $p$ is a homogenous polynomial
such that given any direction $e$, and any scalar $t$
$$
 p(x - t e) \text{ has only real roots}. \tag{1}
$$
This is how many texts on hyperbolic polynomial starts.
The definition above is intuitive, but what are its practical implications?
To give a comparison, in linear programming the objective is down to earth: maximize a linear polynomial over a constrained sets.
With hyperbolic polynomial, the texts generally continue from the definition above
by saying that the function in (1) is convex and they give several other properties.
What is the big picture?
Where does this help in solving an optimization problem?
 A: Apologies for the late answer: I just joined the site, and hyperbolic polynomials in optimization were a particular topic of my graduate studies, so I wanted to make this post my first answer.
The study of hyperbolic polynomials historically originates in the study of PDEs. I can't speak to the intuition there.
But the last paragraph of the OP and the tags on the post indicate that the question, more specifically, is: "[What is the] intuition behind the relevance of hyperbolic polynomials to convex optimization?"
Short answer: because hyperbolic polynomials look like good candidates for an "ideal sweet spot" of both rich theory and practical computational efficiency in convex programming.
Unpacking that involves the following ingredients:


*

*The "scripture" text for modern convex optimization (unless something equally revolutionary has been introduced in since my time of devoted studies in 2009) is Interior-point Polynomial Algorithms in Convex Programming. The profound theoretical breakthrough of Nesterov and Nemirovskii in 1994 was the notion of a self-concordant barrier function. That concept unites hundreds of earlier O.R. theories and papers (including Karmarkar's 1984 ground-breaking paper showing polynomial time complexity for LP). Self-concordant barrier theory is to optimization like Maxwell uniting "electro" with "magnetism" in physics—everything suddenly "fits" together in a beautiful theory that feels to a mathematician's aesthetic sensibilities like it describes "the way things are supposed to be". [Oh wow(!), it just occurred to me in writing this, that the same Maxwell "seems to be the first person to call attention to these [hyperbolic] polynomials" (Güler, 1997)!]

*Every convex set is a slice of a convex cone, and every cone has a well-defined universal self-concordant barrier function. It may have other self-concordant barrier functions as well, but the universal property, allowing the theory to apply to all convex sets, is part of the beauty.

*Despite its theoretical beauty, the universal barrier is not computationally practical in general. It involves an integral over the dual cone.

*That multi-dimensional integral is theoretically tractable in certain special cases. And, again hooray(!), for the most important sub-taxonomies of optimization it corresponds to precisely the same barrier function you would hope for if you studied the best (interior point) algorithms: $\log(x_1 x_2 \ldots x_n)$ for linear programming and $\log \det(M)$ in semidefinite programming ($M$ being the symmetric matrix independant variable being optimized).

*Moreover, in many of those special cases this universal barrier function with closed-form analytic expression also has additional useful properties that make for even nicer theory and faster convergence. So, notably, in LP and SDP your universal self-concordant barrier is also self-scaled (Nesterov and Todd, 1997) allowing for the "best" class of interior point algorithms: long-step primal-dual.

*Self-scaled barriers in fact extend to all symmetric (homogenous and self-dual) cones. In all these cases there is an underlying algebraic structure (Jordan algebras), and the self-scaled barrier corresponds to $\log \det(x)$ where $\det$ is defined by the algebra (constant term of the characteristic polynomial) and provides real eigenvalues (roots of $\det(x-\lambda I)$ for $I$ the algebraic identity element). This algebraic structure provides much of the "leverage" that makes room for the deeper and better theory that can be applied to these special classes of cones. For example, if your barrier is $\log \det(x)$ then the gradient is $x^{-1}$ (the inverse of $x$ in the algebra, which is meaningless in a generic vector space).

*So symmetric cone optimization is in (what I and many others in O.R., at least, would call) a very satisfying theoretical and practical position. Sure, there will always be room for new discoveries, even in LP itself, but the state-of-the-art is really pretty satisfying. The next question (to some of us), then, is: what more can be done to bridge the space between symmetric cone optimization and convex optimization in general? Well, one possible approach to that question (operations researchers are undoubtedly working on many others) is to say, "OK, all symmetric cones have $\log \det(x)$ ($x$ belonging to a Jordan Algebra) for their barrier, how can we generalize the notion of $\det(x)$ while still preserving as many as possible of the nice features that provide for long-step primal-dual interior-point algorithms?"

*Güler (1997) was the first guy (to my knowledge) to make the connection between OR world and PDE world. $\det(x)$ is a multivariate polynomial with guaranteed real roots in a given fixed direction $I$. Optimization needed a generalization of that. Güler told us that folks working on dynamical systems have been studying something like that, not merely for decades, but since the mid-1800's! And guess what(!), $\log p(x)$ is a self-concordant barrier on the hyperbolicity cone corresponding to arbitrary hyperbolic polynomial $p$! And that barrier alone (sans knowledge of any underlying algebraic structure) has proven sufficient qualities to allow long-step interior point algorithms! Hooray! Again, things are starting to feel here like this was a "match made in heaven".

*Yet the primal-dual quality from symmetric cones is not extendable, at least not immediately/directly extendable, because hyperbolicity cones, in general, are not self-dual. Where does that leave a researcher interested in developing the theory? Possibilities include:
(a) Identify an underlying algebraic structure which gives rise to hyperbolic polynomials and cones. For example, the Generalized Lax Conjecture (which is still open, to my knowledge) hypothesizes that every hyperbolicity cone is a linear slice of a symmetric cone, indeed of a real semi-definite cone. That would mean that all hyperbolic programming is in fact reducible to SDP with appropriate subspace constraints. In practice it wouldn't be quite so wonderful as it sounds in theory, because of the growth in the dimensionality of the SDP required to represent a given hyperbolic program, but we would at that point at least be able to say that the classification of difficulty of hyperbolic programming was "satisfying". OR
(b) Even without algebraic structure, milk as much as possible out of the properties of hyperbolic polynomials themselves for which a rich and well-developed theory already exists. Are there yet still more properties from symmetric cones which can be salvaged for usefulness on hyperbolicity cones in general?
In summary, then, the appeal of hyperbolic polynomials in optimization is that $\log p(x)$ is a self-concordant barrier on the hyperbolicity cone which is easily computable and, moreover, "enticingly close/similar to" being a self-scaled barrier. Like any unsolved conjecture, the "enticement" factor here is the yearning to resolve: can we either "connect the dots" or "show that the dots are unconnectable" (and accept hyperbolic programming as inherently "harder" than symmetric programming)? I hope I've answered one OP sub-question: "What is the big picture?"
But (as a theory-leaning guy) I almost forget the other: "Where does this help in solving an optimization problem?" In reality, it doesn't. Last I knew there weren't any real-life hyperbolic programming problems (at the general-level, beyond SDP) waiting to be solved, much less solved with any particular level of efficiency. And in terms of theory, even though Nesterov and Todd discovered self-scaled properties in 1997 without knowledge of Jordan algebras, you wouldn't today want to "tie your hands behind your back" and make life unnecessarily difficult if you were working on solving symmetric cone programming. So depending on how applied vs. pure you are, hyperbolic polynomials arguably don't matter at all for convex optimization, or else they matter a great deal as one of the most promising "missing links".
In any case, whereas Euler's Theorem stood strong and ready when modern cryptography needed it, hyperbolic optimization feels (or "felt", at least to me, as of 2009) "not quite fully ready" if someday called upon as a thing most desirable. Hence, "What is the relevance of hyperbolic polynomials to convex optimization?" is a question which, to my mind, still needs some pieces to fall into place before it can be reckoned as adequately answered. It "seems that" there is more there than we presently know. Resolution of the Generalized Lax Conjecture is one thing which would, IMO, bring a significant degree of closure here.
A: Let me give a little example. In the plane, take $$ f(x,y) = xy.  $$ This is an indefinite quadratic form. When $e = (A,B)$ with both $A,B \neq 0,$ then any line in the plane $\vec{x} + t \vec{e}$ will intersect the pair of axes twice, no matter where the starting point $\vec{x}$ might lie. This includes the case going through the origin with a double root.
However, if $\vec{e} = (1,0)$ with $\vec{x} = (1,1),$ the line  $\vec{x} + t \vec{e}$ intersects the axes just once, at $(0,1).$ Indeed, $f(\vec{x} + t \vec{e}) = f(1+t, 1) = 1+t $ has only one root... need to read something careful about this. There needs to be something about the number of roots staying constant counting multiplicity. 
