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?
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.