A rigidity theorem of \(\alpha \)-relative parabolic hyperspheres (Q1679201)

The authors prove that the only smooth, strictly convex, entire solutions of the equation \[ \text{det} \left( \frac{\partial^2 f}{\partial x_i\partial x_j}\right) = \left(a_{n+1}-\sum a_i\frac{\partial f}{\partial x_i}\right)^\frac{n+2}{\alpha} \;\;\text{ in } \;\;\mathbb{R}^n, \] with \((a_1, a_2, \dots, a_{n+1})\) a constant vector and \(\alpha \notin \left[ \frac{n+2}{n+1}, n+2\right]\), are quadratic polynomials.