  This paper presents a simple, self-contained account of Garding's theory of
hyperbolic polynomials, including a recent convexity result of
Bauschke-Guler-Lewis-Sendov and an inequality of Gurvits. This account also
contains new results, such as the existence of a real analytic arrangement of
the eigenvalue functions.
  In a second, independent part of the paper, the relationship of Garding's
theory to the authors' recent work (arXiv:0710.3991) on the Dirichlet problem
for fully nonlinear partial differential equations is investigated. Let p be a
homogeneous polynomial of degree m on S^2(R^n) which is hyperbolic with respect
to the all positive directions A \geq 0. Then p has an associated eigenvalue
map lambda:S^2(R^n) \to R^m, defined modulo the permutation group acting on
R^m. Consequently, each closed symmetric set E of R^m induces a second-order
p.d.e. by requiring, for a C^2-function u in n-variables, that (D^2 u)(x) lie
in the boundary of E for all x. Assume that E + (R_+)^m is contained in E. A
main result is that for smooth domains in R^n whose boundary is suitably
(p,E)-pseudo-convex, the Dirichlet problem has a unique continuous solution for
all continuous boundary data. This applies to a vast collection of examples the
most basic of which are the m distinct branches of the equation p(D^2 u) =0.
  In the authors' recent extension of results from euclidean domains to domains
in riemannian manifolds (arXiv:0907.1981), a new global ingredient, called a
monotonicity subequation, was introduced. It is shown in this paper that for
every polynomial $p$ as above, the associated Garding cone is a monotonicity
cone for all branches of the the equation p(Hess u) = 0 where Hess u denotes
the riemannian Hessian of u.
