Interior $C^2$ estimate for semi-convex solutions to a class of Hessian quotient equations in arbitrary dimensions

In this paper, we study the interior $C^2$ estimates for Hessian quotient equations $\frac{{\sigma }_3(D^{2}u)}{{\sigma }_l(D^{2}u)}=1$ for $l=1,2$, in arbitrary dimensions, under the natural ellipticity and semi-convexity conditions. We further derive analogous results for the corresponding sum Hessian equations. In addition, we establish several rigidity results.