On Liouville's theorem for the Hessian quotient equation $σ_2/σ_1$

Abstract: We prove Liouville's theorem for semi-convex entire solutions to Hessian quotient equation $σ_2/σ_1=1$ in $\mathbb{R}^n$. The proof is based on the observation that after rewriting the quotient operator as the $σ_2$ operator, acting on a new function, one can refer to the recent result of Shankar and Yuan on Liouville's theorem for $σ_2$ equation.