Local solvability of the $k$-Hessian equations

In this work, we study the existence of local solutions in ${\mathbb{R}}^n$ to $k$-Hessian equation,for which the nonhomogeneous term $f$ is permitted to change the sign or be non negative; if $f$ is $C^{\mathrm{\infty }},$ so is the local solution. We also give a classification for the second order polynomial solutions to the $k-$Hessian equation, it is the basis to construct the local solutions and obtain the uniform ellipticity of the linearized operators at such constructed local solutions.