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

We study interior Hessian estimates for $k$-convex solutions of Hessian quotient equations $\frac{{\sigma }_k(D^{2}u)}{{\sigma }_{\mathrm{\ell }}(D^{2}u)}=1$ with $2\le k\le n$, $0\le \mathrm{\ell }<k$, and $k-\mathrm{\ell }\in \{1,2\}$. We first adopt the unified quantitative separation-propagation method of Li–Wu [LW26b] to derive a doubling inequality, and then use the compactness argument of Shankar–Yuan [SY25] to obtain the estimates. This approach also applies to a class of general Hessian equations.