Interior Hessian estimates for $k$-convex solutions of Hessian quotient equations

We study interior Hessian estimates for $k$-convex solutions of Hessian quotient equations $\frac{σ_k(D^2u)}{σ_\ell(D^2u)}=1$ with $2\leq k\leq n$, $0 \leq \ell < k$, and $k-\ell \in\{1,2\}$. We first adopt the unified quantitative separation-propagation method of Li--Wu [LW26b] to derive a doubling