On Complete Convex Solutions of Equations Similar to the Improper Affine Sphere Equation - Kokarev, V. N. (Журнал математичної фізики, аналізу, геометрії, 2007)
Let $σ_k$ – the sum of all $k$-order Hessian principal minors $(z_{ij})$ for the function $z(x^1,…,x^n)$. If $φ$ of the $n-1$ positive variable belongs to the $C^{3,α}$ class, $0<α<1$ and if it is sufficiently close to the identically single function, then any complete convex solution $z(x^1,…,x^n)$ of the equation $$σ_n=φ(σ_1,…,σ_{n-1})$$ is a quadratic polynomial. Mathematics Subject Classification: 58J05.