On Complete Convex Solutions of Equations Similar to the Improper Affine Sphere Equation

Анотація

Let $\sigma_k$ – the sum of all $k$-order Hessian principal minors $(z_{ij})$ for the function $z(x_1,\ldots,x_n)$. If $\varphi$ of the $n-1$ positive variable belongs to the $C^{3,\alpha}$ class, $0<\alpha<1$ and if it is sufficiently close to the identically single function, then any complete convex solution $z(x_1,\ldots,x_n)$ of the equation
$$\sigma_n=\varphi(\sigma_1,\ldots,\sigma_{n-1})$$ is a quadratic polynomial.

Mathematics Subject Classification: 58J05.