Local solvability of the k Hessian equations
We give a classification of second order polynomial solutions for the homogeneous $k$ Hessian equation \sigma_k[u]=0$ There are only two classes of polynomial solutions: One is convex polynomial; another one must not be (k+1) convex, and in the second case, the $k$ Hessian equations are uniformly elliptic with respect to that solution. Based on this classification, we obtain the existence of $C^\infty$ local solution for nonhomogeneous term $f$ without sign assumptions.