Differential Geometry Seminar
The motivation of defining and study hyperbolic polynomials originates from the theory of PDEs. We call an m-homogeneous polynomial $P(x)$ a-hyperbolic if $P$($ta+x$) has $m$ real roots. Many polynomials are hyperbolic, for instance, the polynomials defined on symmetric matrices: Det($x$), $\sigma_k$($x$) are all Id-hyperbolic. One of Garding’s results is that there exists a convex cone set $C_a$ such that for any $b$ in $C_a, P$ is also hyperbolic for $b$. This fact has importance in the theory of fully nonlinear PDEs. For example, one can verify in which case the $\sigma_k$ equation is elliptic or not.