Interior $$C^2$$ Estimate for Hessian Quotient Equation in General Dimension

In this paper, we study the interior $$C^2$$ regularity problem for the Hessian quotient equation $$\left(\frac{\sigma_n}{\sigma_k}\right)(D^2u)=f$$. We give a complete answer to this longstanding problem: for $$k=n-1,n-2$$, we establish an interior $$C^2$$ estimate; for $$k\leq n-3$$, we show that interior $$C^2$$ estimate fails by finding a singular solution.